Monotonicity Formulas for Capillary Surfaces
Guofang Wang Chao Xia Xuwen Zhang

	
	Monotonicity Formulas]Monotonicity Formulas  for Capillary Surfaces

Wang]Guofang Wang
[G.W]Mathematisches Institut

Universität Freiburg

Ernst-Zermelo-Str.1

79104

Freiburg
 Germany
guofang.wang@math.uni-freiburg.de

Xia]Chao Xia
[C.X]School of Mathematical Sciences

Xiamen University

361005, Xiamen, P.R. China
chaoxia@xmu.edu.cn

Zhang]Xuwen Zhang
[X.Z]Mathematisches Institut

	Universität Freiburg

	Ernst-Zermelo-Str.1

	79104

	Freiburg
 Germany
xuwen.zhang@math.uni-freiburg.de
	

		
§ ABSTRACT 
In this paper, we establish monotonicity formulas for capillary surfaces in the half-space ^3_+ and in the unit ball ^3 and extend the result of Volkmann <cit.> for surfaces with free boundary.
As applications, we obtain Li-Yau-type inequalities  for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in ^3 <cit.>
to the capillary setting,
which is different to another optimal area estimate proved by Brendle in <cit.>.

Keywords: monotonicity formula, Li-Yau inequality, Willmore energy, optimal area estimate, minimal surface, capillary surface 


MSC 2020: 53C42, 53A10, 49Q15 






[
    [
    Received 16 July 2024; accepted 04 September 2024
=====================================================








§ INTRODUCTION

For a 2-dimensional immersed, open surface ⊂^n+1, it is proved by Simon <cit.> that for 0<σ<ρ<∞, a∈ℝ^n+1,

g_a(ρ)-g_a(σ)
=1/π∫_∩B_ρ(a)∖B_σ(a)1/4H⃗+(x-a)^⊥/x-a^2^2^2,

where

g_a(r)   
^2(∩B_r(a))/πr^2+1/16π∫_∩B_r(a)H⃗^2^2
+1/2πr^2∫_∩B_r(a)H⃗·(x-a)^2.

This is known as Simon's monotonicity identity and is later generalized to hold for integral 2-varifolds by Kuwert-Schätzle <cit.>.
As an interesting application, the monotonicity formula yields an alternative proof (by taking ρ∞ and σ0^+) of the Li-Yau inequality (<cit.>):

πΘ_max
≤1/16∫_^n+1H⃗^2μ,

where Θ_max denotes the maximal density of an integral 2-varifold μ, whose generalized mean curvature in ^n+1 is square integrable.
Note that for an immersion of a 2-dimensional compact orientable closed smooth surface F:^n+1,
let μ_g be the induced area measure of  with respect to the pull-back metric g=F^∗ g_ euc, then its image as integral 2-varifold is given by

μF(μ_g)
=(x↦^0(F^-1(x)))^2⌞F().
Hence
from the Li-Yau inequality, we easily see that the Willmore energy

(μ)
1/4∫_^n+1H⃗^2μ≥4π,

and F:^n+1 is an embedding
if

(μ)<8π.

We refer the interested readers to the monographs KS12,MN14 for an overview of the study of Willmore energy.

Recently, Volkmann <cit.> generalized this theory to free boundary surfaces in the unit ball ^n+1={x∈^n+1:x<1} (indeed, he proved it in the context of integral free boundary 2-varifolds).
He obtained a monotonicity identity similar to (<ref>) (see Section <ref>) and consequently a Li-Yau-type inequality.
As an application, he established the Willmore energy for integral free boundary 2-varifolds in ^n+1, and proved that its lower bound is given exactly by 2π.




 §.§ Main Result

In this paper, we deal with the capillary counterpart of the above results.
We first prove a Simon-type monotonicity formula for capillary hypersurfaces in the half-space 
    ^3_+:={x∈^3:x_3<1}
 that are formulated in the weak sense, see Section <ref> for the precise definition.


Given θ∈(0,π), let V be a
rectifiable 2-varifold supported on ^3_+ with its weight measure denoted by μ and let W  a rectifiable 2-varifold supported on ^3_+ with its weight measure denoted by η, satisfying a contact angle condition  as in Definition <ref> (and adopt the notations in Section <ref> below).
For any a∈^3, consider the functions g_a(r),ĝ_a(r) defined by

g_a(r)μ(B_r(a))/πr^2-cosθη(B_r(a))/πr^2+1/16π∫_B_r(a)H⃗^2μ+1/2πr^2∫_B_r(a)H⃗·(x-a)μ
,

ĝ_a(r)μ(B̂_r(a))/πr^2-cosθη(B̂_r(a))/πr^2+1/16π∫_B̂_r(a)H⃗^2μ+1/2πr^2∫_B̂_r(a)H⃗·(x-ã)μ.

Then for any 0<σ<ρ<∞, we have

   (g_a(ρ)+ĝ_a(ρ))-(g_a(σ)+ĝ_a(σ))

   =1/π∫_B_ρ(a)∖B_σ(a)1/4H⃗+(x-a)^⊥/x-a^2^2μ+1/π∫_B̂_ρ(a)∖B̂_σ(a)1/4H⃗+(x-ã)^⊥/x-ã^2^2μ

   -2cosθ∫_B_ρ(a)∖B_σ(a)a_3^2/x-a^4η-2cosθ∫_B̂_ρ(a)∖B̂_σ(a)ã_3^2/x-ã^4η,

where (x-a)^⊥ν_·(x-a)ν_.


As an application, a Li-Yau-type inequality together with the characterization of the equality case is obtained for capillary immersions in the half-space.
For the precise definition of capillary immersions, see Section <ref>. We will adopt the notations that are used in Section <ref>.


The Willmore energy of capillary immersion F is defined in the usual sense, that is,

Given θ∈(0,π) and a θ-capillary immersion F:^3_+.
The Willmore energy of the capillary immersion F is defined as

(F)
1/4∫H⃗^2μ.



 (F) is a conformal invariant with respect to conformal diffeomorphisms. Detailed discussions regarding the definition of the Willmore functional for capillary surfaces are provided in Section <ref>.


Given θ∈(0,π), let F:^3_+ be a compact θ-capillary immersion, satisfying (<ref>).
Then  we have 

    
  * the Li-Yau-type inequality

(F)
≥4(1-cosθ) πΘ^2(μ,x_0)

holds for every x_0∈ F(), where Θ^2(μ,x) denotes the density of μ at x_0.

  * 
the sharp estimate on Willmore energy:    
    
    
    
    

    (F)≥2  (1-cosθ) π,

    
    
    
    
    
    and if 
    (F)<4 (1-cosθ) π,

then F:⊂^3_+ must be an embedding.

  * For θ∈[π/2,π), if 

W(F)<4π,

then F:^3_+ is an embedding.

Moreover, 
equality in (<ref>) holds if and only if F() is a θ-spherical cap.



Inequality (<ref>) was proved using a flow method by the first and the second author in <cit.>*Corollary 1.3 for convex capillary surfaces in ^3_+, while the contact angle θ is restricted within the range (0,π/2].
Therefore Theorem 
<ref> complements the full range of contact angle and moreover removes the assumption of the convexity.

Due to the conformal invariance of the Willmore functional, Theorem <ref>
implies the same inequality as (<ref>) for capillary surfaces in the unit ball, which interestingly implies a new optimal area estimate for minimal capillary surfaces in ^3.


Given θ∈(0,π), let F:^3 be a θ-capillary minimal immersion, satisfying (<ref>).
Then there holds

2-cosθT
≥2(1-cosθ)π.

Equality holds if and only if F() is  a totally geodesic disk in ^3. Here |T| is the area of the “wetting” part defined in Section <ref>.


As a  special case of the isoperimetric inequality for minimal submanifolds proved by Brendle <cit.>, any capillary minimal surface  in the unit ball satisfies the following optimal estimate


≥sin^2θπ=sin^2θ^2

with equality holding if and only if  is a flat disk.
In fact, the higher dimensional counterparts also hold, see <cit.>*Theorem 5.5.
Restricting θ=π/2, i.e., in the free boundary case, this area lower bound was obtained  in <cit.> and <cit.>.
(<ref>) now provides a new optimal area estimate for minimal capillary surfaces in the unit ball. It seems that both area estimates do not imply each other. 
In view of <cit.>, it would be interesting to seek for a higher dimensional generalization of (<ref>).


In the last part of the paper, Section <ref>, we establish a monotonicity formula (<ref>) for capillary
surfaces in ^3. 
This monotonicity formula is  a generalization of Volkmann's theorem  for surfaces with  free boundary in the unit ball <cit.>*Theorem 3.1.
Due to the extra 2-varifold W supported on the unit sphere (as in Theorem <ref>), our monotonicity formula (<ref>) becomes more involved  than his.
We refer the readers to Theorem <ref> below.
Theorem <ref> can also be proved directly by 
this monotonicity formula.




We end the introduction with a short summary of  the fruitful results of the  study of 
capillary surfaces. The existence, regularity, and geometric properties of  capillary surfaces have attracted more and more attention from 
differential geometers and geometric analysts, see the nice book of Finn <cit.> for a through introduction.
Here we mention some recent progress on this topic. 
In ^3, optimal boundary regularity for solutions to the capillarity problems (volume-constrained local minimizers of the free energy functional) was obtained by Taylor <cit.>, while the partial regularity was obtained recently by De Philippis-Maggi DePM15,DePM17 in ^n (n≥4), even in the anisotropic setting.
Quite recently, using the Min-Max method, the existence of capillary minimal or CMC hypersurfaces in compact 3-manifolds with boundary has been shown independently by De Masi-De Philippis <cit.> and Li-Zhou-Zhu <cit.>.
In terms of the classical differential geometry, Hong-Saturnino <cit.> carried out curvature and index estimates for compact and non-compact capillary surfaces.
In <cit.>, we proved a Heintze-Karcher-type inequality and give the characterization of smooth CMC capillary hypersurfaces in the half-space or in a wedge, namely, the Alexandrov-type theorem.
The anisotropic counterpart was tackled in JWXZ23,JWXZ23b as well.
See also <cit.> for a non-smooth generalization.




 §.§ Organization of the Paper

In Section <ref>
we first introduce the capillary surface in the half-space in the setting of varifolds and then
provide the proof of Theorem <ref>, 
In Section <ref> we use the monotonicity formula to estimate the Willmore energy for capillary surfaces in the half-space and prove
the Li-Yau inequality (<ref>) and Theorem <ref>. The Willmore energy for capillary surfaces in  the unit ball is discussed in Section <ref>, together with one proof of Theorem <ref>.
Finally, in Section <ref>, we prove the Simon-type monotonicity formulas in the unit ball, and then provide an alternative proof of Theorem <ref>.





§ MONOTONICITY FORMULA IN THE HALF-SPACE




 §.§ Set-ups

We will be working on the Euclidean space ^3, with  the Euclidean metric  denoted by g_ euc and the corresponding Levi-Civita connection denoted by . ^3_+={x:x_3>0} is the open upper half-space and E_3(0,0,1).

Let ξ:^3^3_+ denote the unique point projection onto the hyperplane ^3_+.
It is easy to see that for any x∈^3, ξ(x)=x-x_3E_3, and hence ξ is a smooth map.
Define

x̃2ξ(x)-x
=x-2x_3E_3

to be the reflection of x across ^3_+.
Given a point a∈^3, define

r=x-a,  r̃=x̃-a,

it is clear  that r̃=x̃-a
=x-ã.

We refer to <cit.> for the background materials in Geometric Measure Theory and consider the following weak formulation of capillary surfaces. 

Given θ∈(0,π), let V be a rectifiable 2-varifold supported on ^3_+ with its weight measure denoted by μ and let W  a rectifiable 2-varifold supported on ^3_+ with its weight measure denoted by η.
(V,W) is said to satisfy a contact angle condition θ if there exists a μ-measurable vector field H⃗∈^1(^3,μ) with H⃗(x)∈ T_x^3_+ for μ-a.e. x∈^3_+,
such that for every X∈ C_c^1(^3;^3) with X

tangent to ^3_+, 

∫_^3div_Xμ-cosθ∫_^3_+div_^3_+Xη=-∫_^3H⃗·Xμ,

where spt μ is countably 2-rectifiable, for simplicity we also set T spt η.
In particular, if X vanishes along ^3_+, there holds that

∫_^3div_Xμ=-∫_^3H⃗·Xμ.



This definition was introduced in <cit.>, which follows from the one initiated in <cit.>,
with the boundary part weakened to be just a 2-varifold.

Notice that the first variation formula (<ref>) is valid if X is merely a Lipschitz vector field.

In this section, we consider those V,W that are integral 2-varifolds with  μ(^3_+)=0
and H⃗∈^2(^3,μ).
It follows that H⃗⊥ for μ-a.e. x∈ thanks to the well known Brakke's perpendicular theorem (<cit.>*Sect. 5.8), so that for any vector v∈^3, one has

21/4H⃗+v^⊥^2
=1/8H⃗^2+2v^⊥^2+H⃗·v,

where v^⊥ denotes the normal part of v with respect to the approximate tangent space T_x.



 §.§ A Monotonicity Formula

A general monotonicity formula for the pairs of varifolds satisfying a contact condition has  been established in <cit.> (see also <cit.>) with none sharp constants, which generalizes a previous result in <cit.> for rectifiable free boundary varifolds.
We are interested in the Simon-type monotonicity formula (<ref>) for the capillary case with optimal constants.


We follow closely the reflection idea in <cit.>. See also <cit.>.
Define l(s)((1/s_σ)^2-1/ρ^2)_+,
where s_σmax{s,σ}.
l(s) is then a Lipschitz cut-off function and it is easy to see that

l(s)
=

    1/σ^2-1/ρ^2,     0<s≤σ,

    1/s^2-1/ρ^2,     σ<s≤ρ,

    0,     ρ<s.


We wish to find a suitable vector field to test (<ref>). The construction is inspired by <cit.> and is nowadays quiet standard.
Precisely,
we define

   X_1(x)=((1/x-a_σ)^2-1/ρ^2)_+(x-a),

   X_2(x)=((1/x-ã_σ)^2-1/ρ^2)_+(x-ã),

   X(x)X_1(x)+X_2(x).

For x∈^3_+, since x=x̃ we simply have

x-a_σ=x̃-a_σ=x-ã_σ,

and hence

X(x)=((1/x-a_σ)^2-1/ρ^2)_+(2x-(a+ã))∈^3_+.


Let B̂_r(a)={x:x-ã<r}
and consider the partitions of ^3:

_1{B_σ(a),B_ρ(a)∖B_σ(a),^3∖B_ρ(a)},

_2{B̂_σ(a),B̂_ρ(a)∖B̂_σ(a),^3∖B̂_ρ(a)}.

We wish to test (<ref>) by X(x). To this end, we compute

∫_Adiv_X_iμ-cosθ∫_Adiv_^3_+X_iη, and ∫_AH⃗·X_iμ
for all sets A∈_i, i=1,2, separately.

For X_2: on 0≤x-ã≤σ, direct computation shows that

X_2(x)=(1/σ^2-1/ρ^2)Id,

and hence

∫_B̂_σ(a)div_X_2μ-cosθ∫_B̂_σ(a)div_^3_+X_2η   =(2/σ^2-2/ρ^2)(μ(B̂_σ(a))-cosθη(B̂_σ(a))),

∫_B̂_σ(a)H⃗·X_2μ   =(1/σ^2-1/ρ^2)∫_B̂_σ(a)H⃗·(x-ã)μ.

On σ<x-ã≤ρ, we have X_2(x)=(1/x-ã^2-1/ρ^2)(x-ã). By  noticing that

(1/x-ã^2)
=-21/x-ã^4(x-ã),

we have

X_2(x)
   =(1/x-ã^2-1/ρ^2)Id-21/x-ã^4(x-ã)⊗(x-ã),

divX_2(x)
   =3(1/x-ã^2-1/ρ^2)-2/x-ã^2
=1/x-ã^2-3/ρ^2,

div_X_2(x)
   =divX_2(x)-X_2[ν_]·ν_=-2/ρ^2+2(x-ã)^⊥/x-ã^2^2,

div_^3_+X_2(x)   =divX_2(x)-X_2[E_3]·E_3
=-2/ρ^2+2ã_3^2/x-ã^4.

It follows that
   
∫_B̂_ρ(a)∖B̂_σ(a)div_X_2μ-cosθ∫_B̂_ρ(a)∖B̂_σ(a)div_^3_+X_2η

=   -2/ρ^2(μ(B̂_ρ(a)∖B̂_σ(a))-cosθη(B̂_ρ(a)∖B̂_σ(a)))
+2∫_B̂_ρ(a)∖B̂_σ(a)(x-ã)^⊥/x-ã^2^2μ

   -2cosθ∫_B̂_ρ(a)∖B̂_σ(a)ã_3^2/x-ã^4η,
   
∫_B̂_ρ(a)∖B̂_σ(a)H⃗·X_2μ=-1/ρ^2∫_B̂_ρ(a)∖B̂_σ(a)H⃗·(x-ã)μ+∫_B̂_ρ(a)∖B̂_σ(a)H⃗·x-ã/x-ã^2μ.

Combining these computations, we obtain
    
∫_^3div_X_2μ-cosθ∫_^3div_^3_+X_2η

=   2/σ^2(μ(B̂_σ(a))-cosθη(B̂_σ(a)))-2/ρ^2(μ(B̂_ρ(a))-cosθη(B̂_ρ(a)))

   +2∫_B̂_ρ(a)∖B̂_σ(a)(x-ã)^⊥/x-ã^2^2μ-2cosθ∫_B̂_ρ(a)∖B̂_σ(a)a_3^2/x-ã^4η,

and

∫_^3H⃗·X_2μ=   1/σ^2∫_B̂_σ(a)H⃗·(x-ã)μ-1/ρ^2∫_B̂_ρ(a)H⃗·(x-ã)μ

   +∫_B̂_ρ(a)∖B̂_σ(a)H⃗·x-ã/x-ã^2μ.

Similar computations hold for X_1 and we obtain
   
∫_^3div_X_1μ-cosθ∫_^3div_^3_+X_1η

=   2/σ^2(μ(B_σ(a))-cosθη( B_σ(a)))-2/ρ^2(μ( B_ρ(a))-cosθη( B_ρ(a)))

   +2∫_ B_ρ(a)∖B_σ(a)(x- a)^⊥/x- a^2^2μ-2cosθ∫_ B_ρ(a)∖B_σ(a)a_3^2/x- a^4η,

and

∫_^3H⃗·X_1μ=   1/σ^2∫_B_σ(a)H⃗·(x- a)μ-1/ρ^2∫_B_ρ(a)H⃗·(x-a)μ

   +∫_ B_ρ(a)∖B_σ(a)H⃗·x-a/x-a^2μ.


On the other hand, thanks to (<ref>), we have







∫_ B_ρ(a)∖B_σ(a)H⃗·x-a/x-a^2μ=2∫_ B_ρ(a)∖B_σ(a)1/4H⃗+(x-a)^⊥/x-a^2^2μ

-1/8∫_ B_ρ(a)∖B_σ(a)H⃗^2μ-2∫_ B_ρ(a)∖B_σ(a)(x-a)^⊥/x-a^2^2μ,

∫_ B̂_ρ(a)∖B̂_σ(a)H⃗·x-ã/x-ã^2μ=2∫_ B̂_ρ(a)∖B̂_σ(a)1/4H⃗+(x-ã)^⊥/x-ã^2^2μ

-1/8∫_ B̂_ρ(a)∖B̂_σ(a)H⃗^2μ-2∫_ B̂_ρ(a)∖B̂_σ(a)(x-ã)^⊥/x-ã^2^2μ.

Putting these facts into the first variation formula (<ref>) and recalling the definition of g_a(r),ĝ_a(r), we deduce (<ref>).






We remark that in Theorem <ref>:
 


  * B_r (a)∩^3_+=B̂_r(a)∩^3_+, and hence η (B_r(a))=η(B̂_r(a)).

  * a_3^2=ã_3^2 and hence the corresponding terms involving them are the same.

    
    



Under the assumptions of Theorem <ref>, the following statements hold:


  * Given θ∈[π/2,π). For every a∈^3, the tilde-density

Θ̃^2(μ-cosθη,a)
lim_r↘0((μ-cosθη)(B_r(a))/πr^2+(μ-cosθη)(B̂_r(a))/πr^2)

exists.
Moreover, the function x↦Θ̃(μ-cosθη,x) is upper semi-continuous in ^3;

  * Given θ∈(0,π).
For every a∈^3, the tilde-density Θ̃^2(μ-cosθη,a) exists, and the function x↦Θ̃(μ-cosθη,x) is upper semi-continuous in ^3_+.







Define

R(r)
   1/2πr^2∫_ B_r(a)H⃗·(x- a)μ+1/2πr^2∫_B̂_r(a)H⃗·(x-ã)μ,



and

G_θ(r)
   (μ-cosθη)(B_r(a))/πr^2+(μ-cosθη)(B̂_r(a))/πr^2

   +1/16π∫_B_r(a)H⃗^2μ+1/16π∫_B̂_r(a)H⃗^2μ+R(r),

then from (<ref>) and thanks to θ∈[π/2,π), we know that G_θ(r) is monotonically nondecreasing, so that
lim_r0^+G_θ(r)
exists.






For R(r), we estimate with Hölder inequality:

R(r)
≤   (μ(B_r(a))/πr^2)^1/2(1/4π∫_B_r(a)H⃗^2μ)^1/2

   +(μ(B̂_r(a))/πr^2)^1/2(1/4π∫_B̂_r(a)H⃗^2μ)^1/2.

Moreover, for 1/4<<1/2, Young's inequality gives

R(r)
≤   μ(B_r(a))/πr^2+1/16π∫_B_r(a)H⃗^2μ

   +μ(B̂_r(a))/πr^2+1/16π∫_B̂_r(a)H⃗^2μ.



By virtue of the monotonicity of G_θ(r), we obtain
    
(μ-cosθη)(B_σ(a))/πσ^2+(μ-cosθη)(B̂_σ(a))/πσ^2

≤   (μ-cosθη)(B_ρ(a))/πρ^2+(μ-cosθη)(B̂_ρ(a))/πρ^2

   +1/16π∫_B_ρ(a)H⃗^2μ+1/16π∫_B̂_ρ(a)H⃗^2μ+R(ρ)+R(σ)

≤   (1+)((μ-cosθη)(B_ρ(a))/πρ^2+(μ-cosθη)(B̂_ρ(a))/πρ^2)

   +1+2^-1/16π(∫_B_ρ(a)H⃗^2μ+∫_B̂_ρ(a)H⃗^2μ)

   +((μ-cosθη)(B_σ(a))/πσ^2+(μ-cosθη)(B̂_σ(a))/πσ^2).


In particular, since 1/4<<1/2, this yields for 0<r<R<∞

   (μ-cosθη)(B_r(a))/πr^2+(μ-cosθη)(B̂_r(a))/πr^2

≤   3((μ-cosθη)(B_R(a))/πR^2+(μ-cosθη)(B̂_R(a))/πR^2)+9/8π∫_^3H⃗^2μ<∞,


which, together with (<ref>), gives

lim_r0^+R(r)=0,

implying that the tilde-density Θ̃(μ-cosθη,a) exists, and also

Θ̃(μ-cosθη,a)
≤((μ-cosθη)(B_ρ(a))/πρ^2+(μ-cosθη)(B̂_ρ(a))/πρ^2)

+1/16π(∫_B_ρ(a)H⃗^2μ+∫_B̂_ρ(a)H⃗^2μ)+R(ρ).

Thus for a fixed R>0 and a sequence of points in ^3 such that x_j a, we have: for 0<ρ<R, it holds that
    
(μ-cosθη)(B_ρ(a))/πρ^2+(μ-cosθη)(B̂_ρ(a))/πρ^2

   ≥lim sup_j∞((μ-cosθη)(B_ρ(x_j))/πρ^2+(μ-cosθη)(B̂_ρ(x_j))/πρ^2)

   ≥lim sup_j∞(Θ̃(μ-cosθη,x_j)-1/16π(∫_B_ρ(x_j)H⃗^2μ+∫_B̂_ρ(x_j)H⃗^2μ)-R(ρ))
   
(<ref>)≥lim sup_j∞Θ̃(μ-cosθη,x_j)

        -C((μ-cosθη)(B_R(a))+(μ-cosθη)(B̂_R(a))/πR^2+(μ))^1/2H⃗_L^2(B_2ρ(a)).

Letting ρ0^+, this gives

Θ̃^2(μ-cosθη,a)
≥lim sup_j∞Θ̃(μ-cosθη,x_j),

which completes the proof of (1).

To prove (2), notice that for any a∈^3_+, a_3=ã_3=0, and hence as θ∈(0,π), the function G_θ(r) is also monotonically nondecreasing for any such a.
Following the proof of (1), we conclude (2).


Suppose that μ,η are compactly supported, we may thus use (<ref>) to find that: for θ∈[π/2,π), a∈^3,

lim_r∞R(r)=0,

and obtain

lim_r∞(g_a(r)+ĝ_a(r))
=1/8π∫_^3H⃗^2μ.

Thus by letting σ0^+ and ρ∞ in (<ref>), we obtain

   1/π∫_^31/4H⃗+(x-a)^⊥/x-a^2^2μ+1/π∫_^31/4H⃗+(x-ã)^⊥/x-ã^2^2μ

   -2cosθ∫_^3a_3^2/x-a^4η-2cosθ∫_^3ã_3^2/x-ã^4η

=   1/8π∫_^3H⃗^2μ-Θ̃^2(μ-cosθη,a).

This is also true for any a∈^3 and any θ∈(0,π/2), provided that Θ̃^2(μ-cosθη,a) exists.

Similarly, for θ∈(0,π) and a∈^3_+, we have

2/π∫_^31/4H⃗+(x-a)^⊥/x-a^2^2μ=1/8π∫_^3H⃗^2μ-Θ̃^2(μ-cosθη,a).






§ WILLMORE ENERGY AND LI-YAU-TYPE INEQUALITY FOR CAPILLARY IMMERSIONS





 §.§ Capillary Immersions



Given a  compact orientable smooth surface , with non-empty boundary .
Let F:^3_+ be an orientation preserving proper immersion, that is, F is smooth on the interior of  and C^2 up to the boundary, such that F( int)⊂^3_+ and F()⊂^3_+. 
In this way, F induces an immersion of  into ^3_+.


Fix a global unit normal field ν to  along F, which determines an orientation on  and an induced orientation on  given by a tangential vector field τ along .
Denote by μ the unit conormal to  in  so that {τ,ν, μ} is compatible with {E_1, E_2, E_3} of ^3. Let ν̅ be the unit normal to  in ^3_+ so that {ν̅, -E_3} compatible with {ν,μ}.

Given θ∈(0,π), we say that the proper immersion F:^3_+ is a capillary immersion (or θ-capillary immersion) with contact angle θ if the angle determined by μ and ν̅ is constant and equals to θ along , that is, for any x∈,

	μ(x)=sinθ(-E_3)+cosθν̅(x).


Abuse of terminology, for the immersion F: →^3_+, we use T to denote the “enclosed domain" by F(), which means the topological boundary T is given by F() and it has induced orientation by τ. We use T to denote the oriented area of T, which is defined to be 
    T=∑_i  sgn(T_i)|T_i|,

where each T_i is a bounded domain such that T_i is a simply closed curve and sgn(T_i)=+1 if ν̅ points outward T_i, while
sgn(T_i)=-1 if ν̅ points inward T_i. |T| is the signed area of the so-called “wetting part”.





Now given  a θ-capillary immersion F:^3_+ for θ∈(0,π)∖{π/2}.


Let g=F^∗ g_ euc be the pull-back metric and μ_g be the induced area measure on .
The induced varifold of  is given by V_g=μ_g⊗_T_p, where T_p is the tangent space  and
its image as varifold is given by V=F_♯(V_g), which is an integral 2-varifold in ^3 with weight measure μ(x↦^0(F^-1(x)))^2⌞ F(), see <cit.>*Section 15.

	
	
	
	
	
	


Define the following measure:

	η   wind(x)^2,

where wind(x) is the winding number of F() about x∈^3. 
Throughout the paper, in terms of capillary immersion, we always assume that

wind(x)≥0 for any x∈^3_+.


With this assumption, η is a positive Radon measure,
and we may relate to it naturally the 2-varifold:

	
	W=η⊗_T_x^3_+.

Let T= sptη⊂^3_+.
By definition of winding number,  the mass (W) of W is exactly the oriented area of T (In fact, sgn(T_i) will be always +1 thanks to (<ref>)).

    The pair (V,W) above satisfies the contact angle condition θ with square integrable generalized mean curvature as in Definition <ref>.


In the case that  is an open surface and F is a properly immersion, the fact that V has square integrable generalized mean curvature in ^3 has already been observed in <cit.>*Section 2.1.
Here we prove the capillary version.

Let us consider any X∈ C_c^1(^3;^3) which is tangent to ^3_+, and let {f_t:^3^3}_t< be the induced variation of X, that is, 

	f_0=id, f_t(^3_+)⊂^3_+,
	 and  /t|_t=0f_t=X,

which induces a family of immersions: {ψ_t}_t<, where ψ_t=f_t∘ F:^3_+.
Ψ:(-,)×^3_+, given by Ψ(t,p)=ψ_t(p), is called an admissible variation.
Let ξ(p)=Ψ/ t(0,p) for p∈, then ξ(p)=X(F(p)).

For this family of immersions, let μ_g_t be the induced area measure of  with respect to the pull-back metric g_t=ψ^*_tg_ euc, the area functional is then given by

	:(-,),  (t)=∫_μ_g_t.

The wetted area functional (t):(-,) is defined by

	(t)
	=∫_[0,t]×Ψ^∗,

where  is the volume form of ^3_+.
We define the free energy functional by

	:(-,),  (t)=(t)-cosθ(t),

from the well-known first variation formula (see e.g., <cit.>*(2.1)) and the capillary condition, we find (recall that ξ(p)=X(F(p)))

	'(0)
	=   -∫_𝐇_(p)<ξ(p),ν(p)>μ_g

	=   -∫_F()∑_p∈F^-1(x)𝐇_(p)<ξ(p),ν(p)>^2(x)

	   -∫<X,𝐇⃗>μ.
	

On the other hand, it is not difficult to see that

	A(t)
	=((f_t)_♯V),

and because (W) is the oriented area,

	'(0)
	=/t|_t=0((f_t)_♯W).

By virtue of the well-known first variation formula for varifolds <cit.>, we get

	
	(V-cosθW)[X]
	='(0)
	=-∫<X,𝐇⃗>μ,

as desired.




We point out that condition (<ref>) is irredundant, because without this constraint, one could simply construct an example as Fig. <ref>, so that we could find some point x∈ F() at which we no longer expect the upper-semi continuity of the tilde density (Proposition <ref>(1)) to be valid.

Precisely, given θ∈(π/2,π), let F:^3_+ be a θ-capillary immersion with F() as in Fig. <ref>.
At the point x, because of the orientation of F(), we have

lim_r0^+η(B_r(x))/πr^2
=-1/2,

and hence

Θ̃^2(μ-cosθη,x)
=1+cosθ<1,

implying that the function ·↦Θ̃(μ-cosθη,·) is not upper semi-continuous at x.











 §.§ Willmore Energy in the Half-Space

The Willmore energy (F) of a smooth immersed compact connected orientable surface F:^3 with boundary , which may have several connected components, is usually proposed by 

	_0(F)
1/4∫_ H_^2μ_g+∫_κ_g,

where κ_g denotes the geodesic curvature of  as a submanifold of .

The reason for proposing (<ref>) is to keep the peculiarity of the Willmore functional—the invariance under the conformal transformation of the ambient space.
Thanks to the well-known Gauß-Bonnet theorem, the Willmore energy may be rewritten as
_0(F)=1/4 ∫_ΣA
^2 +2πχ(),

where A is the traceless second fundamental form and χ() is the Euler characteristic,
it is clear that the first term in (<ref>) is conformal invariant, while the second term is topological invariant. See for example <cit.>.

The point in the above argument is that, in order to keep the conformal invariance, one may indeed add or subtract a topological quantity to the Willmore energy functional.
This insight motivates us to drop the term ∫_κ_g when considering the Willmore energy of the surface  that is capillary immersed into ^3_+, since this term is just a multiple of some topological quantity.
Indeed,
for any θ-capillary immersion F:^3_+, it holds locally

	κ_g
	=_τμ_·τ=_τ(cosθν̅-sinθE_3)·τ=cosθκ̃_g,

where κ̃_g is the geodesic curvature of F() as an immersion in ^3_+,
using the Gauß-Bonnet theorem, one sees

∫_κ_g
=
cosθ∫_κ̃_g
=2πcosθ ind(F()),


where ind(F()) is rotation index of immersed plane curve F(), which is again a topological invariant.

Therefore, in this paper, we regard the Willmore functional for capillary hypersurfaces in ^n+1_+ as 

	(F)    = _0(F)-2πcosθ ind(F())

   =1/4∫_ H_^2μ_g.

W(F) is again a conformal invariant with respect to conformal diffeomorphisms of ^3. 
It matches the Willmore functional for capillary hypersurfaces in the unit ball as well, which will be discussed in the next subsection.


We may rewrite the Willmore energy for the θ-capillary immersion as:

(F)
=1/4∫_H_^2μ_g
=1/4∫H⃗^2μ.









	
	



	
	




The first part of Theorem <ref> amounts to be a direct application of  the monotonicity formula we obtained above. 









    
    
    
    
    

 
    
    
    
    
    
  
   







Let us consider the point x_0∈⊂^3_+.
In this case, x̃_0=x_0 and hence B_r(x_0)=B̂_r(x_0). It follows that
 lim_r↘0^+η(B_r(x_0))+ η( B̂_r(x_0))/πr^2=N(x_0),

for some positive integer N(x_0) =^0(F^-1(x_0)),
since F:^3_+ is an immersion.
Thus we find

Θ̃^2(μ-cosθη,x_0)
=2Θ^2(μ,x_0)-cosθ·N(x_0).

Moreover, since μ is supported on the upper half-space, Θ^2(μ,x_0)=N(x_0)/2.
Hence

Θ̃^2(μ-cosθη,x_0)
=(1-cosθ) N(x_0).


By virtue of the above observation, we infer easily from the monotonicity identity (<ref>) that

(F)
=4∫_^31/4H⃗+(x-x_0)^⊥/x-x_0^2^2μ+4πΘ^2(μ,x_0)-2cosθπN(x_0).

Thus, we obtain the Li-Yau-type inequality:

(F)
≥4(1-cosθ)πΘ^2(μ,x_0)=
2(1-cosθ)πN(x_0),


which implies easily that 

(F)
≥2(1-cosθ) π,






the sharp estimate on the lower bound of Willmore energy.

If W(F)<4(1-cosθ)π, for any x_0∈, the Li-Yau-type inequality (<ref>) implies that N(x_0)=1, and hence F:^3_+ must be an embedding.

Let us now consider the point y_0∈∖. 
Observe that

Θ̃^2(μ-cosθη,y_0)
=Θ^2(μ,y_0).

Letting a=y_0 in (<ref>) and recalling the definition of (F), since θ∈[π/2,π), we obtain readily

2πΘ^2(μ,y_0)
≤(F),  



and hence Θ^2(μ,y_0)=1 if (F)<4π, in which case F:^3_+ must be an embedding. 
This completes the proof.



When F:Σ^3_+ is an embedding, one can infer easily from (<ref>) that

Θ̃^2(μ-cosθη,x_0) =1-cosθ,
 for x_0∈.
Therefore, it may be convenient to define

Θ̃^2(μ-cosθη,x_0)/1-cosθ

as the capillary density of  at a boundary point x_0∈.



A further investigation gives the following characterization of the situation when (F) attains its minima, which also reveals the sharpness of (<ref>).

Given θ∈(0,π), let F:^3_+ be  a capillary immersion.
If (F)=2π(1-cosθ),
then F() must be a θ-cap in ^3_+, i.e., a part of the sphere interesting
^n+1_+ at the angle θ.


We begin by noticing that F:^3_+ is an embedding thanks to Theorem <ref>.
In particular we learn from (<ref>)

that

1/4H⃗(x)+(x-x_0)^⊥/x-x_0^2=0

for μ-a.e. x∈ F() and any x_0∈ F().
It is not difficult to observe that there must be some y∈ F(∖) such that H⃗(y)≠0, otherwise for μ-a.e. x∈ F(), one has (up to a translation, we assume that x_0=0)

x·ν(x)=0,

implying that the enclosing region of F() with ^3_+ is a cone (see e.g. <cit.>*Proposition 28.8), which is not possible.
Consequently, by letting x=y in (<ref>), we obtain for any x_0∈ F() that

x_0-(y-2ν(y)/H⃗(y))^2
=4/H⃗(y)^2,

which shows that F() is indeed a 1-dimensional sphere on ^3_+, say 𝔰.


To proceed, we consider the unique spherical cap in ^3_-, say C_π-θ, which intersects ^3_+ along 𝔰 with the constant angle (π-θ).
Let V_C_π-θ denote the naturally induced varifold of C_π-θ, with weight measure denoted by μ_C_π-θ, which is exactly given by μ_C_π-θ=^2⌞ C_π-θ.
Write μ̃=μ+μ_C_π-θ, = spt(μ̃)=F()∪ C_π-θ.

Since F is a θ-capillary immersion with F:^3_+ an embedding, and because of the construction of C_π-θ,  we see that the integral varifold μ̃ has the following variational structure:
for any X∈ C_c^1(^3;^3),

∫div_ Xμ̃=-∫_X·H⃗μ̃,

where H⃗ is  the generalized mean curvature vector of , which is square integrable on .

Moreover, a direct computation shows that for the spherical cap C_π-θ,

(C_π-θ)
=1/4∫_C_π-θ H⃗^2^2
=2(1-cos(π-θ)
)π=2(1+cosθ)π,

which, together
with (F)=2(1-cosθ)π, yields

(μ̃)
=1/4∫H⃗^2μ̃=4π.

Here (μ̃) is the Willmore energy of μ̃ in ^3, see <cit.>*Appendix A.
It is worth noting that a simple modification of <cit.>*Proposition 4.3 in conjunction with <cit.>*Proposition 2.1.1 shows that for any compactly supported integral 2-varifold with square integrable generalized mean curvature, if its Willmore energy is 4π, then its support is exactly a closed round 2-sphere.



Therefore we conclude that  must be a closed round 2-sphere, and hence F() is a θ-capillary spherical cap as desired.


The last part of the assertion in Theorem <ref>
is simply given by Theorem <ref>.





 §.§ Willmore Energy in the Unit Ball

Due to the conformal invariance of the Willmore functional, the results above can be transferred directly to hold for capillary surfaces in the unit ball.

It is well-known that _0, and hence  defined in (<ref>), are invariant under conformal transformations.
Therefore the Willmore functional for capillary surfaces in the unit ball is the same as ,
i.e.,

(F)
1/4∫_ H⃗^2μ_g+∫_ κ_g-2πcosθ ind(F()) .

Here ind(F()) is the rotation index of the immersed plane curve  ϕ∘ F() where ϕ: ^3∖{p}→^3 is the stereographic projection for some p∉.

Now in this case, the geodesic curvature 
is given by

	κ_g
	=_τμ_·τ=_τ(cosθν̅+sinθN̅(x))·τ=cosθκ̃_g+sinθ,

where N̅(x)=x is the outer unit normal of ^3 and κ̃_g is the geodesic curvature of F:→^2. By the Gauß-Bonnet theorem, we have

2π ind(F())
=∫_κ̃_g +T,

where T is the oriented area of T.
Altogether implying that 

(F)
=1/4∫_ H⃗^2μ_g+
sinθΣ -cosθT.






Given θ∈(0,π), let F:^3 be a θ-capillary immersion.
Then, there holds

(F)
≥2(1-cosθ)π.

Equality holds if and only if F() is a spherical cap or a  totally geodesic disk in ^3.


As explained above, the inequality holds thanks to Theorem <ref> and it suffices to consider the characterization of equality case.
Observe that in the half-space case, equality is achieved if and only if the capillary surfaces are spherical.
Since umbilicity preserves under conformal transformations, we readily see that in the half-ball case, F() should be either a  spherical cap or a totally geodesic disk. 

If θ=π/2, Corollary <ref> was proved in <cit.>.
Now we show how it simply implies Theorem <ref>.





From Corollary <ref>,we find

sinθ-cosθT ≥2(1-cosθ)π.

From div _ x=2 in  we have

2Σ    =∫_x, μ_
=∫_cosθν̅+sinθN̅(x),x

   =sinθ.

(<ref>) follows clearly.







§ MONOTONICITY IDENTITIES IN THE UNIT BALL



 §.§ Set-ups

Let ^3⊂^3 be the Euclidean unit ball centered at the origin, ^2^3 denotes the corresponding unit sphere.

In this section we adopt the following notations:
let ξ:^3∖{0}^3 denote the spherical inversion with respect to ^2.
Fix x_0∈^3 and r>0, B_r(x_0) denotes the open ball of radius r centered at x_0, and

B̂_r(x_0)
=B_r/x_0(ξ(x_0))

denotes the ball of radius r/x_0 centered at ξ(x_0).
A direct computation then shows that: for x∈^2,

x_0x-ξ(x_0)
=x-x_0,

in other words, B_r(x_0)∩^2=B̂_r(x_0)∩^2.

Similar with Definition <ref>, we consider the following weak formulation of capillary surfaces in the unit ball.

Given θ∈(0,π), let V be a rectifiable 2-varifold supported on ^3 with its weight measure denoted by μ, let W be a rectifiable 2-varifold supported on ^2 with its weight measure denoted by η.

(V,W) is said to satisfy the contact angle condition θ if there exists a μ-measurable vector field H⃗∈^1(^3,μ) with H⃗(x)∈ T_x^2 for μ-a.e. x∈^2, such that for every X∈ C_c^1(^3;^3) with X tangent to ^2, it holds that

∫_^3div_Xμ-cosθ∫_^2div_^2Xη=-∫_^3H⃗·Xμ,

where spt(μ) is countably 2-rectifiable.

An important proposition for the pairs of varifolds satisfying contact angle condition is that they have
bounded first variation and satisfy the following first variation formula (see <cit.>*Proposition 3.1).

Given θ∈[π/2,π), let (V,W) be as in Definition <ref>.
Then V-cosθ W has bounded first variation.
More precisely, there exists a positive Radon measure σ_V on ^2 such that

∫_^3div_Xμ-cosθ∫_^2div_^2Xη=-∫_^3H⃗·Xμ

+2∫_^2 X(x)·x(μ-cosθη)+∫_^2X(x)·xσ_V

for every X∈ C_c^1(^3;^3).

If  is in fact a smooth capillary surface embedded in ^3, then we may use a classical computation to see that the above formula is true with σ_V=sinθ^2⌞.
This motivates us to define γ1/sinθσ_V as the generalized boundary measure of V and sptγ as the generalized boundary of V (which are different from the ones obtained from Lebesgue's decomposition theorem).








In all follows, we consider only those V that are integral 2-varifolds with μ(^2)=0 and H⃗∈^2(^3,μ),
it follows that (<ref>) holds.






Moreover, we may rewrite (<ref>) as

∫_^3div_Xμ-cosθ∫_^2div_^2Xη=-∫_^3H⃗·Xμ

-2cosθ∫_^2X(x)·xη+sinθ∫_^2X(x)·xγ
for every X∈ C_c^1(^3;^3).
An immediate consequence is that we may use a compactly supported vector field which coincides with the position vector field in a neighborhood of ^3 to test (<ref>) and obtain

2μ(^3)
=-∫_^3H⃗·xμ+sinθγ(^2).




 §.§ Simon-Type Monotonicity Formulas

Let us first recall that in <cit.>, a Simon-type monotonicity formula is proved for integral 2-varifold μ with μ(^2)=0, that has free boundary in ^3.
Precisely, it is proved that for x_0≠0,
    
1/π∫_B_ρ(x_0)∖B_σ(x_0)1/4H⃗+(x-x_0)^⊥/x-x_0^2^2μ+1/π∫_B̂_ρ(x_0)∖B̂_σ(x_0)1/4H⃗+(x-ξ(x_0))^⊥/x-ξ(x_0)^2^2μ

=   (g_x_0(ρ)+ĝ_x_0(ρ))-(g_x_0(σ)+ĝ_x_0(σ)),

where

g_x_0(r)
μ(B_r(x_0))/πr^2+1/16π∫_B_r(x_0)H⃗^2μ+1/2πr^2∫_B_r(x_0)H⃗·(x-x_0)μ,

ĝ_x_0(r)
g_ξ(x_0)(r/x_0)-x_0^2/πr^2∫_B̂_r(x_0)(x-ξ(x_0)^2+(x-ξ(x_0))^T·x)μ

-x_0^2/2πr^2∫_B̂_r(x_0)H⃗·(x-ξ(x_0)^2x)μ+1/2π∫_B̂_r(x_0)H⃗·xμ+μ(B̂_r(x_0))/π,

and (·)^T denotes the orthogonal projection of a vector onto the approximate tangent space T_x.
For x_0=0, he obtained

1/π∫_B_ρ(0)∖B_σ(0)1/4H⃗+1/πx^⊥/x^2^2μ=(g_0(ρ)+ĝ_0(ρ))-(g_0(σ)+ĝ_0(σ)),

where

g_0(r)
   
μ(B_r(0))/πr^2+1/16π∫_B_r(0)H⃗^2μ+1/2πr^2∫_B_r(0)H⃗·xμ,

ĝ_0(r)   -min(r^-2,1)/2π(2μ(^3)+∫_^3H⃗·xμ).

Volkmann's proof is based on using a properly chosen vector field to test the first variation of a free boundary varifold. This particular choice can be dated back to <cit.>.


We follow to use this vector field to test (<ref>) and obtain the following result.

Given θ∈[π/2,π), let (V,W) be as in Definition <ref>.
Suppose in addition that V is an integral 2-varifold with μ(^2)=0, then it holds that: for x_0≠0, and for every 0<σ<ρ<∞,

   1/π∫_B_ρ(x_0)∖B_σ(x_0)1/4H⃗+(x-x_0)^⊥/x-x_0^2^2μ+1/π∫_B̂_ρ(x_0)∖B̂_σ(x_0)1/4H⃗+(x-ξ(x_0))^⊥/x-ξ(x_0)^2^2μ

   -cosθ/π∫_B_ρ(x_0)∖B_σ(x_0)(x-x_0/x-x_0^2·x)^2η-cosθ/π∫_B̂_ρ(x_0)∖B̂_σ(x_0)(x-ξ(x_0)/x-ξ(x_0)^2·x)^2η

=   (g_x_0,θ(ρ)+ĝ_x_0,θ(ρ))-(g_x_0,θ(σ)+ĝ_x_0,θ(σ)),

where

g_x_0,θ(r)
    g_x_0(r)-cosθη(B_r(x_0))/πr^2,

ĝ_x_,θ(r)
   ĝ_x_0(r)-cosθx_0^2η(B̂_r(x_0))/πr^2

   +cosθx_0^2/πr^2∫_B̂_r(x_0)x-ξ(x_0)^2η-cosθη(B̂_r(x_0))/π.

And for x_0=0,

1/π∫_B_ρ(0)∖B_σ(0)1/4H⃗+1/πx^⊥/x^2^2μ
=(g_0(ρ)+ĝ_0,θ(ρ))-(g_0(σ)+ĝ_0,θ(σ)),

where


ĝ_0,θ(r)   
-min(r^-2,1)/2πsinθγ(^2).



For every 0<σ<ρ<∞, we continue to use the Lipschitz cut-off function l defined in the proof of Theorem <ref>.







    
    
    


For x_0∈^3, let

X_1(x)
l(x-x_0)(x-x_0).


Case 1. x_0≠0.

Define

X_2(x)
   
((1/x-ξ(x_0)_σ/x_0)^2-x_0^2/ρ^2)_+(x-ξ(x_0))

   +(min(x_0x-ξ(x_0),ρ)^2/ρ^2-min(x_0x-ξ(x_0),σ)^2/σ^2)x,

and

XX_1+X_2.

As verified in <cit.>*(7), X is an admissible vector field to test (<ref>) for a.e. 0<σ<ρ<∞.
Moreover, the terms ∫_^3 div_ Xμ and -∫_^3H⃗· Xμ are explicitly computed. That is, if θ=π/2, testing (<ref>) with such X, one gets exactly (<ref>).
Therefore for the case θ∈(π/2,π), it suffices to compute

-cosθ∫_^2div_^2Xη.

To this end, we consider the following decomposition of ^3:

_1{B_σ(x_0),B_ρ(x_0)∖B_σ(x_0),^3∖B_ρ(x_0)},

_2{B̂_σ(x_0),B̂_ρ(x_0)∖B̂_σ(x_0),^3∖B̂_ρ(x_0)},

and we shall compute

-cosθ∫_Adiv_^2X_iη
for all sets A∈_i, i=1,2, separately.






For X_1, a direct computation shows that

X_1
=

(1/σ^2-1/ρ^2)Id,     0≤x-x_0≤σ,

(1/x-x_0^2-1/ρ^2)Id-2x-x_0/x-x_0^2⊗x-x_0/x-x_0^2,     σ<x-x_0≤ρ

0,     ρ<x-x_0,


so that on S^2,

div_^2X_1(x)
=   divX_1(x)-X_1[x]·x

=   

2/σ^2-2/ρ^2,     0≤x-x_0≤σ,

-2/ρ^2+2(x-x_0/x-x_0^2·x)^2,     σ<x-x_0≤ρ,

0     ρ<x-x_0,


and it is easy to deduce that

   -cosθ∫_^2div_^2X_1η

=   -cosθ(2/σ^2η(B_σ(x_0))-2/ρ^2η(B_ρ(x_0))+2∫_B_ρ(x_0)∖B_σ(x_0)(x-x_0/x-x_0^2·x)^2η).


For X_2, we may compute directly to find:

As 0≤x-ξ(x_0)≤σ/x_0,

X_2
=
x_0^2(1/σ^2-1/ρ^2)(x-ξ(x_0))-x_0^2x-ξ(x_0)^2(1/σ^2-1/ρ^2)x;

as σ/x_0<x-ξ(x_0)≤ρ/x_0,

X_2
=(1/x-ξ(x_0)^2-x_0^2/ρ^2)(x-ξ(x_0))-x+x_0^2x-ξ(x_0)^2/ρ^2x;

and X_2≡0 as ρ/x_0<x-ξ(x_0).

Thus X_2 vanishes on ^3∖B̂_ρ(x_0); as 0<x-ξ(x_0)≤σ/x_0,

X_2
=

(1/σ^2-1/ρ^2)x_0^2{(1-x-ξ(x_0)^2)Id-2x⊗(x-ξ(x_0))},





and as σ/x_0≤x-ξ(x_0)≤ρ/x_0,

X_2
=   (1/x-ξ(x_0)^2-x_0^2/ρ^2)Id-2x-ξ(x_0)/x-ξ(x_0)^2⊗x-ξ(x_0)/x-ξ(x_0)^2

   -(1-x_0^2x-ξ(x_0)^2/ρ^2)Id
+2x_0^2/ρ^2x⊗(x-ξ(x_0).

Therefore it is not difficult to compute that: as 0≤x-ξ(x_0)≤σ/x_0,

div_^2X_2(x)
=
2x_0^2(1-x-ξ(x_0)^2)(1/σ^2-1/ρ^2),

and as σ/x_0<x-ξ(x_0)<ρ/x_0,

div_^2X_2(x)
=-2x_0^2(1-x-ξ(x_0)^2)/ρ^2-2+2(x-ξ(x_0)/x-ξ(x_0)^2·x)^2.

It follows that

   -cosθ∫_^2div_^2X_2η

=   -cosθ[(2x_0^2/σ^2η(B̂_σ(x_0))-2x_0^2/σ^2∫_B̂_σ(x_0)x-ξ(x_0)^2η+2η(B̂_σ(x_0)))

   -(2x_0^2/ρ^2η(B̂_ρ(x_0))-2x_0^2/ρ^2∫_B̂_ρ(x_0)x-ξ(x_0)^2η+2η(B̂_ρ(x_0)))

   +2∫_B̂_ρ(x_0)∖B̂_σ(x_0)(x-ξ(x_0)/x-ξ(x_0)^2·x)^2η].

Recall that if θ=π/2, testing (<ref>) with such X, one gets exactly (<ref>).
Now for θ∈(π/2,π), taking (<ref>) and (<ref>) into consideration, we thus obtain (<ref>) for a.e. σ and ρ, and an approximation argument shows that this indeed holds for every σ and ρ.


Case 2. x_0=0.

We continue to use X_1 defined in Case 1 (with x_0=0) and define

X_2(x)
   (min(1,ρ)^2/ρ^2-min(1,σ)^2/σ^2)x,

X
   X_1+X_2.

A direct computation shows that X is an admissible vector field to test (<ref>).
Indeed, we have X=X_1+X_2≡0 on ^2 for every 0<σ<ρ<∞.
Thus

-cosθ∫_^2div_^2Xη=0.

Recall that as θ=π/2, testing (<ref>) with such X one gets exactly (<ref>), therefore for θ∈(π/2,π), we shall get the same identity.
Moreover,
thanks to (<ref>) and invoking the definition of ĝ_0, we may rearrange this and deduce (<ref>) as desired.



Given θ∈[π/2,π). For every x_0∈^3, the tilde-density

   Θ̃^2(μ-cosθη,x_0)

   

lim_r↘0((μ-cosθη(B_r(x_0))/πr^2+(μ-cosθη)(B̂_r(x_0))/πx_0^-2r^2),     x_0≠0,

lim_r↘0μ(B_r(0))/πr^2,     x_0=0,


exists.
Moreover, the function x↦Θ̃(μ-cosθη,x) is upper semi-continuous in ^3.


We prove for the first assertion:

Case 1. x_0≠0.

Define

R_x_0(r)
   1/2πr^2∫_B_r(x_0)H⃗·(x-x_0)μ+x_0^2/2πr^2∫_B̂_r(x_0)H⃗·(x-ξ(x_0))μ

   -x_0^2/πr^2∫_B̂_r(x_0)(x-ξ(x_0)^2+(x-ξ(x_0))^T·x)μ

   -x_0^2/2πr^2∫_B̂_r(x_0)H⃗·(x-ξ(x_0)^2x)μ,

R_x_0,θ(r)
    R_x_0(r)+cosθx_0^2/πr^2∫_B̂_r(x_0)x-ξ(x_0)^2η-cosθη(B̂_r(x_0))/π

   +1/2π∫_B̂_r(x_0)H⃗·xμ+μ(B̂_r(x_0))/π,

then

G_x_0,θ(r)
   (μ-cosθη)(B_r(x_0))/πr^2+x_0^2(μ-cosθη)(B̂_r(x_0))/πr^2

   +1/16π∫_B_r(x_0)H⃗^2μ+1/16π∫_B̂_r(x_0)H⃗^2μ+R_x_0,θ(r)

is monotonically non-decreasing
thanks to (<ref>) and the fact that θ∈[π/2,π).
Therefore

lim_r0^+G_x_0,θ(r)

exists.

Now we estimate with Hölder inequality:

R_x_0,θ(r)
≤   (μ(B_r(x_0))/πr^2)^1/2
(1/4π∫_B_r(x_0)H⃗^2μ)^1/2

   +(μ(B̂_r(x_0))/πx_0^-2r^2)^1/2
(1/4π∫_B̂_r(x_0)H⃗^2μ)^1/2+μ(B̂_r(x_0))/π

   +(μ(B̂_r(x_0))/πx_0^-2r^2)^1/2(μ(B̂_r(x_0))/π)^1/2+(μ(B̂_r(x_0))/π)^1/2(1/4π∫_B̂_r(x_0)H⃗^2μ)^1/2

   -2cosθη(B̂_r(x_0))/π+1/2π∫_B̂_r(x_0)H⃗μ+μ(B̂_r(x_0))/π,

where we have used the fact that sptμ⊂^3 so that x≤1.
Moreover, for 1/8<<1/4, Young's inequality gives

R_x_0,θ(r)
≤   μ(B_r(x_0))/πr^2+1/16π∫_B_r(x_0)H⃗^2μ

   +μ(B̂_r(x_0))/πx_0^-2 r^2+1/16π∫_B̂_r(x_0)H⃗^2μ

   +μ(B̂_r(x_0))/πx_0^-2r^2+1/4μ(B̂_r(x_0))/π+μ(B̂_r(x_0))/4π+1/4π∫_B̂_r(x_0)H⃗^2μ

   +2(μ-cosθη)(B̂_r(x_0))/π+1/2π∫_B̂_r(x_0)H⃗μ

≤   2(μ(B_r(x_0))/πr^2+μ(B̂_r(x_0))/πx_0^-2r^2)+^-1/16π(∫_B_r(x_0)∪B̂_r(x_0)H⃗^2μ)

   +11+^-1/4π(μ-cosθη)(B̂_r(x_0))+3/8π∫_B̂_r(x_0)H⃗^2μ,

where we have used again θ∈[π/2,π) and the fact that

1/2π∫_B̂_r(x_0)H⃗μ≤μ(B̂_r(x_0))/2π+1/8π∫_B̂_r(x_0)H⃗^2μ.


By virtue of the monotonicity of G_x_0,θ(r), we obtain for 0<σ<ρ<∞
   
(μ-cosθη)(B_σ(x_0))/πσ^2+(μ-cosθη)(B̂_σ(x_0))/πx_0^-2σ^2

≤   (μ-cosθη)(B_σ(x_0))/πσ^2+(μ-cosθη)(B̂_σ(x_0))/πx_0^-2σ^2

   +1/16π∫_B_ρ(x_0)H⃗^2μ+1/16π∫_B̂_ρ(x_0)H⃗^2μ+R_x_0,θ(ρ)+R_x_0,θ(σ)

≤   (1+2)((μ-cosθη)(B_ρ(x_0))/πρ^2+(μ-cosθη)(B̂_ρ(x_0))/πx_0^-2ρ^2)

   +1+2^-1/16π∫_B_ρ(x_0)∪B̂_r(x_0)H⃗^2μ+11+^-1/2π(μ-cosθη)(^3)

   +3/4π∫_^3H⃗^2μ+2((μ-cosθη)(B_σ(x_0))/πσ^2+(μ-cosθη)(B̂_σ(x_0))/πx_0^-2σ^2).

In particular, since 1/8≤≤1/4, this yields for a fixed R>0 that

   (μ-cosθη)(B_r(x_0))/πr^2+(μ-cosθη)(B̂_r(x_0))/πx_0^-2r^2

≤   3((μ-cosθη)(B_R(x_0))/πR^2+(μ-cosθη)(B̂_R(x_0))/πx_0^-2R^2)

   +23/4π∫_^3H⃗^2μ+19/π(μ-cosθη)(^3)<∞
for every 0<r<R.

This, in conjunction with (<ref>), yields

lim_r0^+R_x_0,θ(r)=0,

and also implies that the tilde-density Θ̃(μ-cosθη,x_0) exists. Consequently

Θ̃(μ-cosθη,x_0)
≤((μ-cosθη)(B_r(x_0))/πr^2+(μ-cosθη)(B̂_r(x_0))/πx_0^-2r^2)

+1/16π∫_B_r(x_0)H⃗^2μ+1/16π∫_B̂_r(x_0)H⃗^2μ+R_θ,x_0(ρ).














Case 2. x_0=0.

Define

R_0,θ(r)
1/2πr^2∫_B_r(0)H⃗·xμ,

then

G_0,θ(r)
μ(B_r(0))/πr^2+1/16π∫_B_r(0)H⃗^2μ-min(r^-2,1)/2πsinθγ(^2)
+R_0,θ(r)

is monotonically non-decreasing thanks to (<ref>), thus

lim_r0^+G_0,θ(r)

exists.

Using Hölder inequality and Young's inequality, we obtain for 1/4≤≤1/2,

R_0,θ(r)
≤   (μ(B_r(0))/πr^2)^1/2(1/4π∫_B_r(0)H⃗^2μ)^1/2

≤   μ(B_r(0))/πr^2+1/16π∫_B_r(0)H⃗^2μ.


By virtue of the monotonicity of G_0,θ(r), we obtain for 0<σ<ρ≤1

μ(B_σ(0))/πσ^2
≤   μ(B_ρ(0))/πρ^2+1/16π∫_B_ρ(0)H⃗^2μ+R_0,θ(ρ)+R_0,θ(σ)

≤   (1+)μ(B_ρ(0))/πρ^2+1+2^-1/16π∫_B_ρ(0)H⃗^2μ+μ(B_σ(0))/πσ^2,

where we have used that ĝ_0,θ(r)≡-sinθ/2πγ(^2) for every 0<r≤1 in the first inequality.
In particular, since 1/4≤≤1/2 this yields for any 0<r<1:

μ(B_r(0))/πr^2
≤3μ(B_1(0))/π+9/8π∫_^3H⃗^2μ<∞,

which, together with (<ref>), yields

lim_r0^+R_0,θ(r)=0,

and also implies that the tilde-density Θ̃(μ-cosθη,0) exists.
Consequently for 0<r<1,

Θ̃(μ,0)
≤μ(B_r(0))/πr^2+1/16π∫_B_r(0)H⃗^2μ+R_0,θ(r).


Now we verify the upper semi-continuity by definition,
for a sequence of points x_j x_0 and for 0<ρ<1/2, we have

   (μ-cosθ)(B_ρ(x_0))/πρ^2+(μ-cosθ)(B̂_ρ(x_0))/πx_0^-2ρ^2

≥   lim sup_j∞((μ-cosθ)(B_ρ(x_j))/πρ^2+(μ-cosθ)(B̂_ρ(x_j))/πx_j^-2ρ^2)

≥   lim sup_j∞(Θ̃(μ-cosθη,x_j)-1/16π(∫_B_ρ(x_j)H⃗^2μ+∫_B̂_ρ(x_j)H⃗^2μ)-R_x_j,θ(ρ))

≥   lim sup_j∞Θ̃(μ-cosθη,x_j)-C((μ-cosθη)(B_1/2(x_0)∪B̂_1/2(x_0))/π(1/2)^2

   +∫_^3H⃗^2μ+(μ-cosθη)(^3))
·(H⃗_L^2(B_2ρ(x_0))+(μ(B̂_ρ(x_0))/π)^1/2)

   -2(μ-cosθη)(B̂_ρ(x_0))/π,

where we have used (<ref>), (<ref>) for the last inequality, here we interpret B̂_r(0)=∅ and (μ-cosθ)(B̂_ρ(0))/π0^-2ρ^2=0.
Letting ρ0^+, this gives

Θ̃(μ-cosθη,x_0)≥lim sup_j∞Θ̃(μ-cosθη,x_j),

which completes the proof of the second assertion.


Since sptμ and sptη are compact, we see from the definition that

   lim_r∞R_x_0,θ(r)=1/2π∫_^3H⃗·xμ+(μ-cosθη)(^3)/π,  x_0≠0,

   lim_r∞R_0,θ(r)=0,

and obtain from the Simon-type monotonicity formulas (letting σ0^+,ρ∞)
    
1/π∫_^31/4H⃗+(x-x_0)^⊥/x-x_0^2^2+1/4H⃗+(x-ξ(x_0))^⊥/x-ξ(x_0)^2^2μ

   -cosθ/π∫_^3(x-x_0/x-x_0^2·x)^2+(x-ξ(x_0)/x-ξ(x_0)^2·x)^2η

=   lim_ρ∞G_x_0,θ(ρ)-lim_σ0^+G_x_0,θ(σ)

=   1/8π∫_^3H⃗^2μ+1/2π∫_^3H⃗·xμ+(μ-cosθη)(^3)/π-Θ̃(μ-cosθη,x_0)

=   1/8π∫_^3H⃗^2μ+sinθγ(^2)-2cosθη(^3)/2π-Θ̃(μ-cosθη,x_0)

for x_0≠0, and for x_0=0:

1/π∫_^31/4H⃗+x^⊥/x^2^2μ
=1/16π∫_^3H⃗^2μ+sinθγ(^2)/2π-Θ̃(μ,0).






 §.§ Applications of Eqn. <ref>


As a by-product of the monotonicity formula, we may establish the lower bound for the Willmore functional, which is conformal invariant and was proved in the previous Section. Moreover, we may obtain the optimal area estimate for minimal capillary surfaces in the unit ball in a rather direct way, which needs not go through the
Willmore functional.



In the case that x_0∈^2, we clearly have ξ(x_0)=x_0.
Besides, for any x∈^2, there holds x-x_0^4=4(1-(x· x_0))^2, and hence a direct computation shows that

(x-x_0/x-x_0^2·x)^2+(x-ξ(x_0)/x-ξ(x_0)^2·x)^2
=1/2 on ^2.

Taking this into account, the Simon-type monotonicity formula (<ref>) then reads

1/π∫_^31/4H⃗+(x-x_0)^⊥/x-x_0^2^2+1/4H⃗+(x-ξ(x_0))^⊥/x-ξ(x_0)^2^2μ

=1/8π∫_^3H⃗^2μ+sinθγ(^2)-cosθη(^3)/2π-Θ̃(μ-cosθη,x_0).

Now since F: Σ→^3 be a capillary minimal immersion, and also recall (<ref>), we obtain 

1/π∫_^3(x-x_0)^⊥/x-x_0^2^2+(x-ξ(x_0))^⊥/x-ξ(x_0)^2^2μ=2-cosθT/2π-Θ̃(μ-cosθη,x_0).

As (<ref>) we have Θ̃(μ-cosθη,x_0)=(1-cosθ) N(x_0), where N(x_0) = ^0(F^-1(x_0)).
Theorem <ref> then follows.    
 








alpha


