Some Inequalities for the Omori-Yau Maximum Principle

Abstract

We generalize A. Borbély’s condition for the conclusion of the Omori-Yau maximum principle for the Laplace operator on a complete Riemannian manifold to a second-order linear semielliptic operator with bounded coefficients and no zeroth order term. Also, we consider a new sufficient condition for the existence of a tamed exhaustion function. From these results, we may remark that the existence of a tamed exhaustion function is more general than the hypotheses in the version of the Omori-Yau maximum principle that was given by A. Ratto, M. Rigoli, and A. G. Setti.

1. Introduction

Let be a smooth complete Riemannian manifold of dimension . For a smooth real-valued function on , a second-order linear differential operator without zeroth-order term can be written as where is self-adjoint with respect to , is the Hessian of in the form defined by for , and finally . In this paper, we will deal with the semielliptic case, that is, is positive semidefinite at each point, and we always assume that

Definition 1. A smooth complete Riemannian manifold is said to satisfy the Omori-Yau maximum principle for the Laplace operator (the above semielliptic operator ) if for any function which is bounded from above and for any there is a point such that , , and ().

The Omori-Yau maximum principle is a useful substitute of the usual maximum principle in noncompact settings. For the operator , Definition 1 is the well-known Omori-Yau maximum principle for the Laplacian, which was first proven by Omori [1] and Yau [2] when the Ricci curvature is bounded below. This was improved upon by Chen and Xin [3] and Ratto et al. [4] when the Ricci curvature decays were slower than a certain decreasing function tending to minus infinity. For instance, we have the following.

Theorem 2 (Ratto-Rigoli-Setti’s condition [4, Theorem  2.3]). Let be a fixed point and be the distance function from . Let one assumes that away from the cut locus of one has where is some constant and on satisfies Then satisfies the Omori-Yau maximum principle for the Laplacian .

Borbély [5, Theorem] has given an elegant proof of the validity of the Omori-Yau maximum principle where the Ricci curvature condition (3) is replaced by the assumption without (4) and (5). Also, Bessa et al. [6, Theorem ] proved Borbély’s theorem [5, Theorem] for the -Laplacian for a selected smooth function on . In this paper, we first show that Borbély’s theorem [5, Theorem] is also true for our semielliptic operator by following his method in [5] (see Theorem 5).

To state other results, we need the following definitions.

Definition 3. Let be a real-valued continuous function on and let a point . (i)A function is called proper, if the set is compact for every real number .(ii)A function defined on a neighborhood of is called an upper-supporting function for at , if the conditions and hold in .

Definition 4. A proper continuous function is called a -tamed exhaustion, if the following condition holds:(1).(2)At all points it has a smooth, upper-supporting function at defined on an open neighborhood such that and .

Royden [7] showed that every complete Riemannian manifold satisfying Omori-Yau’s condition (i.e., the Ricci curvature is bounded from below) admits a -tamed exhaustion function. Inspired by Royden’s article [7], Kim and Lee [8, Theorem ] proved the Omori-Yau maximum principle for the Laplacian when there exists a -tamed exhaustion function. Moreover, they proved that every complete Riemannian manifold satisfying Ratto-Rigoli-Setti’s condition admits a -tamed exhaustion function [8]. Similar to Definition 4, we define an -tamed exhaustion function (i.e., we replace with ) [9, Definition ]. Then, using the existence of an -tamed exhaustion function, Hong and Sung [9, Theorem ] generalized the Omori-Yau maximum principle for the Laplacian to the operator . In this paper, we give a new sufficient condition for the existence of an -tamed exhaustion function (see Theorem 6). We prove this result using the ideas adapted from [8]. Note that Theorem 6, together with [9, Theorem ], implies the maximum principle of Omori and Yau for the operator . As a corollary, we prove that the existence of a -tamed exhaustion is more general than Ratto-Rigoli-Setti’s condition. Unfortunately, for the operator , the relation between Borbély’s condition (or the existence of an -tamed exhaustion) and Ratto-Rigoli-Setti’s condition remains for further study.

Now, we formulate our main results. From (1), is diagonalizable at each point on an orthonormal basis, since is symmetric. Then one can take a normal coordinate around such that at is represented as a diagonal matrix. Thus, we have for a real-valued function on , where each is nonnegative; the entries and are bounded above as varies by (2). We introduce a locally defined differential operator for convenience as follows: Put and for . We may assume that and are the largest of and , respectively.

Then we have the following.

Theorem 5. Let be a fixed point and be the distance function from . Assume that for all where is smooth, , and on satisfies Then satisfies the Omori-Yau maximum principle for the operator .

Theorem 6. Let be a fixed point and be the distance function from . Assume that for all where is smooth, , and on satisfies Then admits an -tamed exhaustion function.

Remark 7. By [5, Corollary] and Theorem 6, Ratto-Rigoli-Setti’s condition without implies the existence of a -tamed exhaustion function. Therefore, the existence of a -tamed exhaustion function for the conclusion of the Omori-Yau maximum principle for the Laplacian is more general than the hypothesis in Theorem 2.

There are some other sufficient conditions under which the Omori-Yau maximum principle for the Laplacian holds [10–12]. Also, [13] deals with the general setting of semielliptic operators (trace type operators). Recently, Bessa and Pessoa [14, Theorem ] present a sufficient condition for the conclusion of the Omori-Yau maximum principle for a second-order linear semielliptic operator with bounded first-order coefficients and no zeroth-order term. However, they will not consider the existence of a tamed exhaustion function as sufficient conditions for the conclusion of the Omori-Yau maximum principle.

2. Proof of Theorem 5

The proof is similar to the method in [5]. Let . We may assume that at every point of ; otherwise, has its maximum at some point and that point directly satisfies the Omori-Yau maximum principle for a semielliptic operator .

Define the function as Then Since on , we have , and . Hence the function is strictly increasing, and . Since the set is compact, we have For any positive constant , we define the function as Then Because, for all , and . If , then we have Define as Then, clearly, . Furthermore, we can obtain for all ; that is, there is a point such that . Assume that to the contrary for all . Then we will show that there is a constant with such that for all . This is a contradiction to the definition of .

Let . Because , there is a sufficiently large positive number such that for . Also, because the set is compact, the statement for all implies that there is a constant with such that for . Now, let . Then, for , we have for all . Moreover, by (17) and , we have .

Next, we have to show that is smooth at . Since , it is enough to show that is smooth at . To avoid confusion, the point , in the statement of Theorem 5, is switched to . Note that is a Lipschitz function and is smooth on , where is the cut locus of . Suppose that . Then we have two possibilities (Petersen [15, Lemma ]); either there are two distinct minimizing geodesic segments joining to , or there is a geodesic segment from to along which is conjugate to . Notice that We consider the first case. Let and . Since and are distinct segments, we have . For or , the functions are differentiable on and they have a left-derivative at . Note that is smooth on . From the definition of , , and we obtain where denotes the directional derivative of at the point in the direction of . Furthermore, since has a directional derivative at in the direction of , we have This yields Hence, by (21) and (23), we get the following inequality: Note that and . Recall that . Then, from (24), we can get The inequality (25) will lead to a contradiction. Since and are different segments, by connecting from the point to the point with a geodesic segment, there is a constant with such that, for a sufficiently small , the distance . Thus there is a constant with depending only on the angle of and such that for a sufficiently small . Note that . By plugging (26) to (25), we have a contradiction.

From now, let us consider the second case. Since is distance minimizing between and , is smooth at for . Let . Then is also smooth for . Because is conjugate to along , by a simple calculation, we get Because , by (23), we get ; that is, . Hence the level surface is a smooth hypersurface near . Denote by the surface parallel to and passing through the point for some . Since is smooth near , the surface is also smooth near for a sufficiently small . Therefore, by (27), for some sufficiently small , the trace of the second fundamental form of at in the direction of is greater than , where is the trace of the second fundamental form of the geodesic sphere at with respect to the normal vector . This implies that there has to be a point sufficiently close to , which lies inside ; that is, Since is parallel to , we also have a point on such that the distance . By (28), we have Since is strictly increasing, we get This is a contradiction to the fact that for all . Therefore, the function must be smooth at .

By the definition of , , , and , we have Because , , and , we have Hence Recall notations (6) and (7). Since we have Note that . By (31), (33), and , the first equality of (35) yields Also, by (2), (31), (33), (36), , and , the second inequality of (35) yields If we replace with , then the above inequality, (32), and (36) show that the point satisfies the conditions in Definition 1.

3. Proof of Theorem 6

The proof is similar to the method in [8]. Let be a fixed point and be the distance function from . Define a function by Assume that a smooth complete Riemannian manifold satisfies assumption (10). Then we will prove that is an -tamed exhaustion function. We consider two cases.

First Case. Assume that has no cut points in .

By the definition, the function is an exhaustion function for . We have to show that, for certain positive constants and , and outside a ball of a certain radius with center . Let and . Then . By a direct calculation, one gets By (12), there is a positive constant such that Then, for , we obtain Moreover, by (11), we have By plugging (41) to (39), we have Note that . Applying (42) gives By (2) and (44), one gets By assumption (11), we have Because of the above inequality, , (41), and (42), we have for By our assumption (10), there exits such that Thus, by (45) and (48), we have If we replace with , then satisfies the additional conditions for an -tamed exhaustion function.

Second Case. Assume that the cut locus of is nonempty.

Let be a cut point of and let for . We choose a point outside of cut locus of such that and . Denote by . Take such that and does not have cut point of .

Now, we present several functions to find an upper-supporting function for .

For a neighborhood , we define a smooth map with , and it is translation sending to in a coordinate chart including both and and satisfying . Also, we define a function such that , , , and Since and , we get . Finally, for , we define a function where and for . Note that we choose as close to such that . Therefore, .

Let . Then one gets . Because of the fact and the inequality (41), we get Moreover, we have two inequalities; that is, for , Hence is an upper-supporting function for at the point .

Since , , , and , we have By our assumption (2), the above inequality implies that Notice that where . By a simple calculation, we have and hence Using , , (52), (56), and (58), we have Let be the distance to a closest cut point of . Because the point is a cut point of , by (41) and (42), we get By plugging (62) to (60), our assumption (10) tells us that, for , Therefore, by (55) and (63), we obtain, for , So satisfies the conditions for an -tamed exhaustion function.

Altogether, we can conclude that must be an -tamed exhaustion function for .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the referee for valuable comments and corrections. Also, the author thanks Professor G. P. Bessa for pointing out [6, 14].