

Maps from Riemannian manifolds into non-degenerate Euclidean cones

Luciano Mari    Marco Rigoli
111Mathematic subject classification 2000: primary 53C42, 35B50; secondary 53C21.  Keywords: maximum principles, harmonic maps, isometric immersion, Riemannian manifold.

Let be a connected, non-compact -dimensional Riemannian manifold. In this paper we consider smooth maps with images inside a non-degenerate cone. Under quite general assumptions on , we provide a lower bound for the width of the cone in terms of the energy and the tension of and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case is an isometric immersion, we also show that, if is sufficiently well-behaved and has non-positive sectional curvature, cannot be contained into a non-degenerate cone of .

Dipartimento di Matematica, Università degli studi di Milano,

Via Saldini 50, I-20133 Milano (Italy)

E-mail addresses: ,

1 Introduction

In a famous paper Omori, [O], studied minimal immersions of manifolds into cones of Euclidean space and in the process he proved (a version of) what is by now known as the Omori-Yau maximum principle. This powerful tool has since then been applied to solve several geometric problems from Yau’s pioneering work on the generalized Schwarz lemma, to the study of the group of conformal diffeomorphisms of a manifold, [PRS2], to the theory of submanifolds (see also the very recent papers [APD] and [AG]) and so on; for a short account one can consult [PRS1] and the references therein. The Omori-Yau principle can be briefly stated in the following form:

Let be a complete Riemannian manifold and let be a function satisfying . Then, under appropriate geometric assumptions on the manifold there exists a sequence with the following properties:

for every .

It turns out that in many applications property plays no role. This fact suggests to ignore it and to look for possibly lighter geometric assumptions to ensure the validity of conclusions and . Indeed, in [PRS2] the authors show that the sole assumption of stochastic completeness of (that is, the property of the Brownian motion on to have infinite intrinsic lifetime) is equivalent to the simultaneous validity of and for every , , on an appropriate sequence.
To distinguish between the two situations we shall refer to this new statement as to the weak maximum principle. These two principles are indeed different. For instance, one immediately realizes that stochastic completeness does not even imply geodesic completeness (as an example, consider the standard punctured Euclidean space ).

As a matter of fact, geometric applications often lead to consider differential inequalities of the form

for some , . This and a number of other considerations have suggested to generalize the known results on stochastic completeness and the weak maximum principle to a wider class of elliptic operators. In particular, to symmetric diffusion operators on , where are positive smooth functions on . Theorem 3.10 of [PRS1] states the equivalence between -stochastic completeness (here means that the diffusion having infinite intrinsic lifetime is that generated by ) and the weak maximum principle for . For a detailed account, one can consult [PRS1], [RSV], where the authors deal even with the case .
Together with diffusions, one can also consider the full Hessian operator. In this case we state the following definition:

The weak maximum principle for the Hessian operator holds on if for every , , there exists a sequence such that


It is worth to observe, in view of applications, that -stochastic completeness and the weak maximum principle for the Hessian can be granted via the existence of appropriate proper functions defined in a neighborhood of infinity in . This is achieved by the use of Khas’minskii test (see [PRS1], where for convenience the result is stated for the Laplacian only) and of its generalization in the case of the Hessian (see again [PRS1] and also question in [PRS4]). Observe that adding the condition to (1.1) gives the well known Omori-Yau maximum principle for the Hessian.

In this paper we shall make use of Theorem (and subsequent remarks), Theorem 3.10 and Proposition of [PRS1] to prove the geometrical results that we are now going to describe.

Let , , be the -dimensional Euclidean space endowed with its canonical flat metric. Fix an origin and a unit vector . We set , shortly , to denote the non-degenerate cone with vertex in , direction and width , , that is,


Let be a connected, -dimensional Riemannian manifold (), and let

be a smooth map. We indicate with the square of the Hilbert-Schmidt norm of the differential (in other words, twice the energy density of ) and with the tension field of . In case is an isometric immersion, and , where is the mean curvature vector. We fix an origin and we consider the distance function from , . We set for the geodesic ball with radius centered at .

Since the work of Omori, the study of maps with image contained in a cone has captured the attention of researchers. Indeed, important progresses in the understanding of their geometry have been made by a number of authors and in particular, regarding the content of the present paper, by [BK], [PRS1] and by [A], where through stochastic methods it is proved that if is stochastically complete and is an harmonic map with energy density bounded from below away from zero, cannot be contained into any non-degenerate cone of . However, all these references limit themselves to consider the harmonic (or minimal) case.

On the contrary, the two main results in this paper aim at determining a lower bound for the width of the cone containing when is merely a smooth map or an isometric immersion, where of course we expect the estimate to depend on the position of in the cone.

To state the next theorems, given , we define


The constant can be easily computed, but the actual value is irrelevant for our purposes. Note also that is non-decreasing as a function of .

Theorem 1.4.

Let M be a connected, non-compact -dimensional Riemannian manifold, and let

be a map of class such that on . Consider the elliptic operator , and assume that is -stochastically complete. Let be a cone with vertex at , let be the hyperplane orthogonal to passing through and let be the Euclidean distance between this hyperplane and .

If is contained in , then


In case is an isometric immersion, we can replace with in (1.5) obtaining a sharper estimate.

Remark 1.

Note that, in case

and , that is, ”gathers around the origin ”, as we shall see in the proof, we have no restriction on .

Remark 2.

The condition that generates a conservative diffusion (i.e. -stochastic completeness) is implied by

where the integral is evaluated along the paths of the Brownian motion starting from , and is the intrinsic lifetime of . As in the case of -stochastic completeness, no geodesic completeness is required. If is complete, -stochastic completeness can be obtained using generalized versions of a previous work of Grigor’Yan [G], see [S] and [PRS1] (Theorem 3.15 and subsequent propositions). In particular, if there exist , such that




then the weak maximum principle for holds. It is worth to observe that (1.7) implies , but no restriction on nonnegativity of is needed. In case , an application of [S] leads to slightly improving (1.7) to

The converse of the above result is in general false; indeed, it is well known that, on a radially symmetric model in the sense of Greene and Wu, [GW], -stochastic completeness is equivalent to the condition


which is implied by (1.7) with but not equivalent. We remark, in passing, that it is an open problem to establish whether or not (1.8) is sufficient for -stochastic completeness on a generic complete manifold.

Remark 3 (Sharpness of inequality 1.5).

Due to the form of (1.5), we cannot expect the result to be significant when is far from , in the following sense: for every , and satisfying the assumptions of Theorem 1.4, and for every , we can consider the map . Then , while the other parameters in the RHS of (1.5) remain unchanged. Therefore, for sufficiently large inequality (1.5) becomes meaningless unless . On the contrary, we show with a simple example that, when is very small, (1.5) is sharp in the following sense: for every fixed hyperplane , and for every origin , there exists a family of maps , representing , such that, if we denote with the width of the non-degenerate tangent cone containing ,

for some constant . Indeed, for every fixed consider the hypersurface given by the graph , with the induced metric. Indicating with the hyperplane , we have by standard calculations

Therefore . Moreover, for the tangent cone passing through the origin

thus, since , we reach the desired conclusion.

As an immediate consequence of Theorem 1.4, we recover Atsuji’s result [A]:

Corollary 1.9.

Let be harmonic and such that for some positive constant . If is stochastically complete, then cannot be contained in any non-degenerate cone of . In particular, a stochastically complete manifold cannot be minimally immersed into a non-degenerate cone of .

Note that even the statement of [A] in its full generality requires , an assumption that can be overcome by the weighted requests (1.6), (1.7). Furthermore, in case we can replace -stochastic completeness and the uniform control from below in (1.6) with the properness of .

Corollary 1.10.

Let be a Riemannian manifold. Then, there does not exist any proper harmonic map , such that on and is contained into a non-degenerate cone of .

Remark 4.

It is a well known open problem to deal with the case , that is, when the cone degenerates to a half-space and the dimension is greater than . When , , by Hoffman-Meeks’ half-space Theorem [HM] the only properly embedded minimal surfaces in a half-space are affine planes. On the contrary, if there exist properly embedded minimal hypersurfaces even contained between two parallel hyperplanes (the so called generalized catenoids). It is still an open problem to find sufficient conditions on in order to have a Hoffman-Meeks’ type result, and it seems quite difficult to adapt the methods of the proof of (1.4) for this purpose.

The next application of Theorem 1.4 has a topological flavor. This result, which is interesting when is not proper, ensures that some kind of ”patological” gathering around points of does not occur when the map is sufficiently well behaved. To make the corollary more transparent, we state it using the sufficient conditions (1.6) and (1.7). First we introduce the following

Definition 1.11.

Let be a convex subset of . A point is called an -corner of if it is the vertex of a non-degenerate cone containing .

Corollary 1.12.

Let be a complete Riemannian manifold and let be a map of class . Suppose that (1.6) holds, and that


for some and as in (1.6). Assume also that (1.7) holds. Then, the convex envelope contains no -corners.

The second main theorem of this paper is obtained with a modification of the idea used in the proof of Theorem 1.4. We obtain

Theorem 1.14.

Let be an isometric immersion of an -dimensional manifold satisfying the weak maximum principle for the Hessian into a non-degenerate cone . Assume the codimension restriction


and suppose that the sectional curvature of satisfy


for some constant . Then


where is as in (1.3).

Remark 5.

With the notation for some function on , we mean that, for every and every -plane , the sectional curvature of in satisfy .

As a consequence, we get the following corollaries: the former generalizes results of Tompkins [T], Chern-Kuiper [CK] and Jorge-Koutroufiotis [JK], whereas the latter improves on [D], Theorem .

Corollary 1.18.

Let be a complete -dimensional Riemannian manifold with sectional curvature satisfying


for some , and where stands for the -iterated logarithm. Then, cannot be isometrically immersed into a non-degenerate cone of .

Corollary 1.20.

Let be a Kähler manifold such that the weak maximum principle for the Hessian holds. Then cannot be isometrically immersed into a non-degenerate cone of .

Remark 6.

The inequality on the left of (1.19) provides a sharp sufficient condition for the full Omori-Yau maximum principle for the Hessian. Indeed, it would be sufficient that this inequality hold along -planes containing (see Theorem 1.9 of [PRS1]). As far as we know, it is an open problem to obtain other general sufficient conditions ensuring the validity of the weak maximum principle for the Hessian.

2 Proof of the Theorems

2.1 Proof of Theorem 1.4

First of all we observe that

and that the right hand side of (1.5) is invariant under homothetic transformations of . We choose as the origin of global coordinates, and for the ease of notation we set

Furthermore, for future use, note that implies


Next, we reason by contradiction and we suppose that (1.5) is false. Therefore, there exists such that

By definition, and the fact that the inequality is strict, we can find

such that



For the ease of notation we set and ; the last relation becomes


Note also that


Now, we define the following function:

and we note that, by construction, . We first claim that


indeed, an algebraic manipulation shows that (2.5) is equivalent to

On the other hand, using (2.1) the LHS of the above inequality is bounded from below by since , and the claim is proved.

We now consider the closed non empty set:

Using (2.1) and the definition of we deduce:


Since by construction, we can square inequality (2.6) to obtain


Since , the LHS of the above inequality is a quadratic polynomial in with two distinct roots (use Cartesio rule and (2.5)), where are given by

therefore, (2.7) implies


We then use the elementary inequality on to deduce

and thus (2.8) together with yields


To compute , we fix a local orthonormal frame and its dual coframe . Then, writing , a simple computation shows that


and taking the covariant derivative we have , where

Tracing the above expression we get


on , where we have defined


Note that, by (2.1),


and that


The possibility, for the isometric case, of substituting with in (1.5) depends only on the above difference. Since the next passages are the same, we carry on with the general case. Substituting (2.13), (2.14) in (2.11) it follows that


We now restrict our estimates on the RHS of (2.15) on . Then, (2.9) holds and from (2.3) we obtain

Inserting this inequality into (2.15) we have


We want to find a strictly positive lower bound for on . Since represents a convex parabola and since is increasing in the variable on , its maximum is attained either in or in . Since , we have

thus we can roughly bound as follows:

and the RHS of the above inequality is strictly positive since . Therefore, (2.16) together with (2.9) yield


for some .
There are now two possibilities:

  • is an absolute maximum for on . By assumption , and the finite form of the maximum principle yields , so that . Since (2.17) immediately gives the contradiction.

  • . In this case, since is bounded above on , it is enough to evaluate inequality (2.17) along a sequence realizing the weak maximum principle for , that is , . Note that this sequence eventually lies in .

This concludes the proof.

2.2 Proof of Corollary 1.9

If is stochastically complete and , then it is straightforward to deduce that is -stochastically complete, where . Indeed, for every with , along the sequence realizing the weak maximum principle for we have also

The result follows setting in Theorem 1.4.

2.3 Proof of Corollary 1.10

From (2.9) in the proof of Theorem 1.4 we deduce that is bounded, hence is compact. The properness assumption implies that is compact, thus is compact. Therefore, it is enough to use the finite form of the maximum principle in (2.17).

2.4 Proof of Corollary 1.12

We reason by contradiction and let be an -corner.

  • If fix a small ball around contained in , and cut the corner transversally with an hyperplane sufficiently near to ; it is immediate to see that in this way we produce a convex set containing and strictly smaller than , contradiction.

  • Suppose now , and let such that . Consider the map ; by assumption, there exists a direction such that , thus by continuity we can take a curve

    with small such that on . Therefore, is an immersed curve in , and this fact contradicts the assumption that is an -corner.

  • If , choose as the hyperplane orthogonal to the direction of the cone and passing through . It follows that . By (1.6) and (1.13), we argue that is bounded above on . By Remark 2, (1.6) and (1.7) ensure that is -stochastically complete, where . By Theorem 1.4 we conclude the validity of (1.5) which gives , contradiction.

2.5 Proof of Theorem 1.14

We follow the proof of Theorem 1.4 verbatim substituting (2.2) with


for some , (2.3) with


and arrive up to inequality (2.9) included. Next, we fix and we let be orthonormal vectors spanning the -plane . From Gauss equations and (1.16) we have


where is the second fundamental form. Since , by Otsuki lemma (see [D], Lemma ) it follows that there exists a unit vector such that

hence, from (2.19) and (2.9) we deduce


Next, we let , , be the geodesic characterized by the initial data

Call the arc-length parameter and define the function

A simple computation, using the fact that is an isometric immersion, gives:


where has the expression in (2.12), evaluated at . Since

Setting , evaluating at we deduce


Inserting (2.21) into (2.23) we get


Proceeding as in the proof of Theorem 1.4, since

where is independent from and from .
On the other hand, a standard computation using the fact that is a geodesic and the definition of the Hessian of a function, gives . Putting together the last two inequalities we obtain


If is an absolute maximum of , then from (2.25) we immediately contradict the finite maximum principle, otherwise


and (2.25) gives


contradicting the validity of the weak maximum principle for the Hessian operator since the function in bounded above on . This completes the proof of Theorem 1.14.

2.6 Proof of Corollary 1.18

By (1.19), using Theorem of [PRS1] we have the validity of the weak maximum principle for the Hessian. The result follows immediately setting and .

2.7 Proof of Corollary 1.20

The proof follows the same lines as in [D], so we only sketch it. From the assumptions, since the codimension is , for every the theory of flat bilinear forms ensure the existence of a vector , such that . We define as in Theorem 1.14. Expression (2.22) gives at every point , and for every ,

This calculation is independent from the value of . If is chosen to be sufficiently small that (note that, by definition, on it holds and ), evaluating along a sequence satisfying the weak maximum principle for the Hessian we deduce, for sufficiently large,

for every . This fact contradicts the existence of .

3 Ending remarks

We end the paper with some observations regading Theorem 1.14 and Corollary 1.18. In particular, we stress the difference of applicability between the weak maximum principles for the Laplacian and for the full Hessian operator. A first striking difference is pointed out by Proposition 40 of [PRS4], which states that every Riemannian manifold satisfying the weak maximum principle for the Hessian must be non-extendible (that is, non isometric to any proper open subset of another Riemannian manifold). For example, for every Riemannian manifold and , does not satisfy the weak maximum principle for the Hessian. This is in sharp contrast to Theorem IX. of [C], whose immediate application is the following

Proposition 3.1.

Let be a stochastically complete -dimensional manifold with Riemannian measure , and let be a compact submanifold of dimension such that is connected. Then, if , is stochastically complete.

Denoting with and with the Dirichlet heat kernel on , by Theorem IX. of [C] uniformly on compact subsets of . Therefore, the Dirichlet heat kernel of coincides with , and since the Brownian motion on satisfies

for every , , and this shows that is stochastically complete.

For those familiar with stochastic calculus and potential theory this proposition may look almost trivial since it turns out that, in these assumptions, the set is polar with respect to the Brownian motion (i.e. it has zero capacity).
Since, by Theorem 1.9 of [PRS1], geodesic completeness and a well-behaved sectional curvature imply the full Omori-Yau maximum principle for the Hessian, one might ask if, keeping a well-behaved sectional curvature and relaxing geodesic completeness to the property of non-extendibility, one could prove the validity of the weak maximum principle for the Hessian. This is false, as the following simple counterexample shows. Consider on Euclidean space the standard cone

In polar coordinates , where and , the cone can be parametrized as