# Some Inequalities for the Omori-Yau Maximum Principle

Authors: Kyusik Hong
Journal: Abstract and Applied Analysis
 (2015)

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## 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 operatorL 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.


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## Body

## 1. Introduction

Let(M,g) be a smooth complete Riemannian manifold of dimension n. For a smooth real-valued function h on M, a second-order linear differential operator L:C∞(M)→C∞(M) without zeroth-order term can be written as(1)Lh=Tr⁡A∘Hessh+gV,∇h,where A∈Γ(End(TM)) is self-adjoint with respect to g, Hessh∈Γ(End(TM)) is the Hessian of h in the form defined by Hessh(X)=∇X∇h for X∈Γ(TM), and finally V∈Γ(TM). In this paper, we will deal with the semielliptic case, that is, A is positive semidefinite at each point, and we always assume that(2)supM⁡Tr⁡A+supM⁡V<∞.Definition 1.
A smooth complete Riemannian manifoldM is said to satisfy the Omori-Yau maximum principle for the Laplace operator Δ (the above semielliptic operator L) if for any C2 function h:M→R which is bounded from above and for any ϵ>0 there is a point xϵ∈M such that hxϵ-supMh<ϵ, ∇h(xϵ)<ϵ, and Δh(xϵ)<ϵ (Lh(xϵ)<ϵ).The Omori-Yau maximum principle is a useful substitute of the usual maximum principle in noncompact settings. For the operatorΔ, 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]).
Leto∈M be a fixed point and r(x) be the distance function from o. Let one assumes that away from the cut locus of o one has(3)Ricc∇r,∇r≥-n-1BG2r,where B>0 is some constant and G(t) on [0,∞) satisfies(4)∫0∞1Gtdt=∞,G0=1,G′≥0,G2k+10=0,∀k≥0,(5)limsupt→∞⁡tGtGt<∞.Then M satisfies the Omori-Yau maximum principle for the Laplacian Δ.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 Δr(x)≤G(r(x)) without (4) and (5). Also, Bessa et al. [6, Theorem 5.6] proved Borbély’s theorem [5, Theorem] for the f-Laplacian Δf for a selected smooth function on M. In this paper, we first show that Borbély’s theorem [5, Theorem] is also true for our semielliptic operator L by following his method in [5] (see Theorem 5).To state other results, we need the following definitions.Definition 3.
Letu be a real-valued continuous function on M and let a point p∈M. (i)
A functionu is called proper, if the set {p:u(p)≤r} is compact for every real number r.(ii)
A functionv defined on a neighborhood Up of p is called an upper-supporting function for u at p, if the conditions v(p)=u(p) and v≥u hold in Up.Definition 4.
A proper continuous functionu:M→R is called a Δ-tamed exhaustion, if the following condition holds:(1)
u≥0.(2)
At all pointsp∈M it has a C2 smooth, upper-supporting function v at p defined on an open neighborhood Up such that ∇v|p≤1 and Δv|p≤1.Royden [7] showed that every complete Riemannian manifold satisfying Omori-Yau’s condition (i.e., the Ricci curvature is bounded from below) admits a Δ-tamed exhaustion function. Inspired by Royden’s article [7], Kim and Lee [8, Theorem 2] proved the Omori-Yau maximum principle for the Laplacian Δ when there exists a Δ-tamed exhaustion function. Moreover, they proved that every complete Riemannian manifold satisfying Ratto-Rigoli-Setti’s condition admits a Δ-tamed exhaustion function [8]. Similar to Definition 4, we define an L-tamed exhaustion function (i.e., we replace Δ with L) [9, Definition 1.4]. Then, using the existence of an L-tamed exhaustion function, Hong and Sung [9, Theorem 2.1] generalized the Omori-Yau maximum principle for the Laplacian Δ to the operator L. In this paper, we give a new sufficient condition for the existence of an L-tamed exhaustion function (see Theorem 6). We prove this result using the ideas adapted from [8]. Note that Theorem 6, together with [9, Theorem 2.1], implies the maximum principle of Omori and Yau for the operator L. As a corollary, we prove that the existence of a Δ-tamed exhaustion is more general than Ratto-Rigoli-Setti’s condition. Unfortunately, for the operator L, the relation between Borbély’s condition (or the existence of an L-tamed exhaustion) and Ratto-Rigoli-Setti’s condition remains for further study.Now, we formulate our main results. From (1), A is diagonalizable at each point on an orthonormal basis, since A is symmetric. Then one can take a normal coordinate (x1,…,xn) around xϵ∈M such that A at xϵ is represented as a diagonal matrix. Thus, we have(6)Lhxϵ=∑lallxϵ∂2∂xl2hxϵ+∑lalxϵ∂∂xlhxϵ,for a real-valued function h on M, where each all(xϵ) is nonnegative; the entries all(xϵ) and |al(xϵ)| are bounded above as xϵ varies by (2). We introduce a locally defined differential operator for convenience as follows:(7)Δ~xϵ≔a11xϵ∂2∂x12+⋯+annxϵ∂2∂xn2,∇~xϵ1≔a1xϵ∂∂x1+⋯+anxϵ∂∂xn,∇~xϵ≔a11xϵ∂∂x1,…,annxϵ∂∂xn.Put dl=all(xϵ) and el=|al(xϵ)| for 1≤l≤n. We may assume that d1 and e1 are the largest of {d1,…,dn} and {e1,…,en}, respectively.Then we have the following.Theorem 5.
Leto∈M be a fixed point and r(x) be the distance function from o. Assume that for all x∈M(8)Δ~xrx≤Grx,where r is smooth, r(x)>1, and G(t) on [0,∞) satisfies (9)∫0∞dtGt=∞,G≥1,G′≥0.Then M satisfies the Omori-Yau maximum principle for the operator L.Theorem 6.
Leto∈M be a fixed point and r(x) be the distance function from o. Assume that for all x∈M(10)Δ~xrx≤Grx,where r is smooth, r(x)>1, and G(t) on [0,∞) satisfies(11)∫0∞dtGt=∞,G≥1,G′≥0,(12)limsupt→+∞⁡tGtGt<+∞.Then M admits an L-tamed exhaustion function.Remark 7.
By [5, Corollary] and Theorem 6, Ratto-Rigoli-Setti’s condition without G2k+1(0)=0∀k≥0 implies the existence of a Δ-tamed exhaustion function. Therefore, the existence of a Δ-tamed exhaustion function for the conclusion of the Omori-Yau maximum principle for the Laplacian Δ is more general than the hypothesis in Theorem 2.There are some other sufficient conditions under which the Omori-Yau maximum principle for the LaplacianΔ holds [10–12]. Also, [13] deals with the general setting of semielliptic operators (trace type operators). Recently, Bessa and Pessoa [14, Theorem 1] 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 Theorem5

The proof is similar to the method in [5]. Let U=sup⁡h. We may assume that h<U at every point of M; otherwise, h has its maximum at some point and that point directly satisfies the Omori-Yau maximum principle for a semielliptic operator L.Define the functionF(t) as(13)Ft=e∫0t1/Gsds.Then(14)F′=FG.Since G≥1 on [0,∞), we have F≥1, and F′>0. Hence the function F is strictly increasing, and limt→∞F(t)=∞. Since the set {x∈M:r(x)≤1} is compact, we have (15)U-sup⁡hx:rx≤1>0.For any positive constant ϵ<min{1,U-sup{h(x):r(x)≤1}}, we define the function hλ:M→R as (16)hλx=λFrx+U-ϵ.Then(17)hλx>hxifrx≤1,λ≥0.Because, for all x∈M, F(r(x))≥1 and U>h(x). If λ>ϵ, then we have(18)hλx>hx,∀x∈M.Define λ0 as (19)λ0=inf⁡λ:hλx>hx,∀x∈M.Then, clearly, λ0>0. Furthermore, we can obtain hλ0(x)≥h(x) for all x∈M; that is, there is a point xϵ∈M such that hλ0(xϵ)=h(xϵ). Assume that to the contrary hλ0(x)>h(x) for all x∈M. Then we will show that there is a constant λ′ with λ0>λ′ such that hλ′(x)>h(x) for all x∈M. This is a contradiction to the definition of λ0.Letλ0>λ1. Because limr→∞F(r)=∞, there is a sufficiently large positive number r0 such that hλ1(x)>U>h(x) for r(x)>r0. Also, because the set {x∈M:r(x)≤r0} is compact, the statement hλ0(x)>h(x) for all x∈M implies that there is a constant λ2 with λ0>λ2 such that hλ2(x)>h(x) for r(x)≤r0. Now, let λ′=max{λ1,λ2}. Then, for λ0>λ′, we have hλ′(x)>h(x) for all x∈M. Moreover, by (17) and λ0>0, we have r(xϵ)>1.Next, we have to show thathλ0 is smooth at xϵ. Since hλ(x)=λF(r(x))+U-ϵ, it is enough to show that r is smooth at xϵ. To avoid confusion, the point o, in the statement of Theorem 5, is switched to p. Note that r is a Lipschitz function and is smooth on M∖{p,Cp}, where Cp is the cut locus of p. Suppose that xϵ∈Cp. Then we have two possibilities (Petersen [15, Lemma 8.2]); either there are two distinct minimizing geodesic segments γ1,γ2:[0,t0]→M joining p to xϵ, or there is a geodesic segment γ:[0,t0]→M from p to xϵ along which xϵ is conjugate to p. Notice that(20)t0=rγit0=rxϵfori=1or2.We consider the first case. Let w=γ1′(t0) and v=γ2′(t0). Since γ1 and γ2 are distinct segments, we have w≠v. For i=1 or 2, the functions t→r(γi(t)) are differentiable on (0,t0) and they have a left-derivative at t0. Note that h is C2 smooth on M. From the definition of λ0, hλ0≥h, and hλ0(xϵ)=h(xϵ) we obtain(21)liminfs→0+⁡hλ0γ2t0+s-hλ0γ2t0s≥Dvhxϵ,where Dvh(xϵ) denotes the directional derivative of h at the point xϵ in the direction of v. Furthermore, since hλ0 has a directional derivative at xϵ in the direction of -v, we have(22)-λ0F′t0=-λ0F′rxϵ=D-vhλ0xϵ≥D-vhxϵ=-Dvhxϵ.This yields(23)Dvhxϵ≥λ0F′rxϵ.Hence, by (21) and (23), we get the following inequality:(24)liminfs→0+⁡hλ0γ2t0+s-hλ0γ2t0s≥λ0F′rxϵ.Note that hλ0γ2′=λ0F′rγ2r′(γ2) and r(γ2(t0))=r(xϵ). Recall that λ0>0. Then, from (24), we can get(25)liminfs→0+⁡rγ2t0+s-rγ2t0s≥1.The inequality (25) will lead to a contradiction. Since γ1 and γ2 are different segments, by connecting from the point γ1(t0-s) to the point γ2(t0+s) with a geodesic segment, there is a constant c with 0<c<1 such that, for a sufficiently small s>0, the distance d(γ1(t0-s),γ2(t0+s))<c2s. Thus there is a constant c′ with 0<c′<1 depending only on the angle of v and w such that(26)rγ2t0+s<t0+c′s,for a sufficiently small s>0. Note that r(γ2(t0))=t0. By plugging (26) to (25), we have a contradiction.From now, let us consider the second case. Sinceγ is distance minimizing between p and xϵ, r is smooth at γ(t) for 0<t<t0. Let m(t)=Δr(γ(t)). Then m(t) is also smooth for 0<t<t0. Because γ(t0) is conjugate to p=γ(0) along γ, by a simple calculation, we get(27)limt→t0-⁡mt=-∞.Because λ0F′(r(xϵ))>0, by (23), we get Dvh(xϵ)>0; that is, ∇h(xϵ)≠0. Hence the level surface H={x∈M:h(x)=h(xϵ)} is a C2 smooth hypersurface near xϵ. Denote by Hs the surface parallel to H and passing through the point γ(t0-s) for some s>0. Since H is C2 smooth near xϵ, the surface Hs is also C2 smooth near γ(t0-s) for a sufficiently small s>0. Therefore, by (27), for some sufficiently small s, the trace of the second fundamental form of Hs at γ(t0-s) in the direction of γ′(t0-s) is greater than m(t0-s), where m(t0-s) is the trace of the second fundamental form of the geodesic sphere B(p,t0-s) at γ(t0-s) with respect to the normal vector γ′(t0-s). This implies that there has to be a point qs∈Hs sufficiently close to γ(t0-s), which lies inside B(p,t0-s); that is,(28)rqs<t0-s.Since Hs is parallel to H, we also have a point on q∈H such that the distance d(qs,q)=s. By (28), we have (29)rq<t0=rxϵ.Since F is strictly increasing, we get (30)hλ0q=λ0Frq+U-ϵ<λ0Frxϵ+U-ϵ=hλ0xϵ=hxϵ=hq.This is a contradiction to the fact that hλ0(x)≥h(x) for all x∈M. Therefore, the function r must be smooth at xϵ.By the definition ofF, F≥1, G≥1, and G′≥0, we have(31)0<F′=FG,F′′=F′G-FG′G2=FG2-FG′G2≤FG2.Because λ0>0, F≥1, and h(xϵ)=λ0F(r(xϵ))+U-ϵ<U, we have(32)0<-λ0Frxϵ+ϵ=U-hxϵ<ϵ.Hence(33)λ0<ϵFrxϵ≤ϵ.Recall notations (6) and (7). Since(34)hλ0x≥hx,∀x∈M,hλ0xϵ=hxϵ,we have(35)∇hλ0xϵ=∇hxϵ,Lhλ0xϵ≥Lhxϵ.Note that ∇r=1. By (31), (33), and G≥1, the first equality of (35) yields(36)∇hxϵ=λ0F′rxϵ∇rxϵ<ϵFrxϵFrxϵGrxϵ≤ϵ.Also, by (2), (31), (33), (36), G≥1, and Δ~xϵr≤G, the second inequality of (35) yields(37)Lhxϵ≤Lhλ0xϵ=∑lallxϵ∂2∂xl2hλ0xϵ+∑lalxϵ∂∂xlhλ0xϵ≤λ0F′rxϵΔ~xϵrxϵ+F′′rxϵ∇~xϵrxϵ·∇rxϵ+e1ϵ<ϵFrxϵFrxϵGrxϵGrxϵ+d1FrxϵGrxϵ2+e1ϵ≤ϵ1+d1+e1.If we replace ϵ with ϵ(1+d1+e1), then the above inequality, (32), and (36) show that the point xϵ satisfies the conditions in Definition 1.


## 3. Proof of Theorem6

The proof is similar to the method in [8]. Let o∈M be a fixed point and r(x) be the distance function from o. Define a function u:M→R by (38)ux=∫0rx2Gs-1ds.Assume that a smooth complete Riemannian manifold satisfies assumption (10). Then we will prove that u is an L-tamed exhaustion function. We consider two cases.First Case. Assume that o has no cut points in M.By the definition, the functionu is an exhaustion function for M. We have to show that, for certain positive constants C and C1, ∇u<C and Lu<C1 outside a ball of a certain radius with center xϵ. Let ϕ(t)=exp{∫0t‍G(s)-1ds} and B(xϵ,r)={x∈M∣dist(x,xϵ)<r}. Then u(x)=log⁡ϕ(r(x)2). By a direct calculation, one gets(39)∇u=∇log⁡ϕr2=2r∇rϕ′r2ϕr2=2r∇rGr2-1.By (12), there is a positive constant C such that(40)r2GrGr2=r2GrGr2-1<C4.Then, for r>1, we obtain(41)rGrGr2-1<r2GrGr2-1<C4.Moreover, by (11), we have(42)sup0,∞⁡Gr-1=inf0,∞⁡Gr-1≤1.By plugging (41) to (39), we have(43)∇u<12∇rCGr-1.Note that ∇r=1. Applying (42) gives(44)∇u<C2.By (2) and (44), one gets(45)∇~xϵ1u<e1C2.By assumption (11), we have(46)ϕ′r2ϕr2′=Gr2-1′=-Gr2-2G′r2≤0.Because of the above inequality, ∇~xϵr≤d1, (41), and (42), we have for r>1(47)Δ~xϵu=Δ~xϵlog⁡ϕr2=4r2ϕ′r2ϕr2′∇~xϵr2+2Gr2-1∇~xϵr2+rΔ~xϵr≤2Gr2-1∇~xϵr2+rΔ~xϵr≤2rGr2-1d12r-1+Δ~xϵr<C2Gr-1d12r-1+Δ~xϵr<C2d12+C2Gr-1Δ~xϵr.By our assumption (10), there exits r0>1 such that(48)Δ~xϵu<C2d12+C2onM∖Bxϵ,r0.Thus, by (45) and (48), we have (49)Lu=Δ~xϵu+∇~xϵ1u<C2d12+1+e1onM∖Bxϵ,r0.If we replace C/2(d12+1+e1) with C1, then u satisfies the additional conditions for an L-tamed exhaustion function.Second Case. Assume that the cut locus of o is nonempty.Letxϵ be a cut point of o and let F(t)=log⁡ϕ(t2) for t>0. We choose a point xϵ^ outside of cut locus of o such that dist(xϵ,xϵ^)<1 and r(xϵ^)>r(xϵ). Denote by B(y,r)={x∈M∣dist(x,y)<r}. Take η,δ>0 such that B(xϵ,η)∩B(xϵ^,δ)=∅ and B(xϵ^,δ) does not have cut point of o.Now, we present several functions to find an upper-supporting function foru.For a neighborhoodU⊂B(xϵ,η), we define a smooth map T:U→B(xϵ^,δ) with Txϵ(xϵ)=xϵ^, and it is translation sending xϵ to xϵ^ in a coordinate chart including both B(xϵ,η) and B(xϵ^,δ) and satisfying r(T(x))≥r(x). Also, we define a C2 function λ such that λ(xϵ)=1, ∇λ(xϵ)=0, Δλ(xϵ)=0, and (50)λxrTx≥rx+rxϵ^-rxϵonU.Since r(xϵ^)>r(xϵ) and r≥0, we get λ(x)>0. Finally, for x∈U, we define a function(51)Hx=Nx+12F′′rxϵλxrTx-rxϵ^2whenF′′rxϵ>0,Nx-12F′′rxϵ^rTx-rxϵ^2whenF′′rxϵ<0,Nx+12QrxϵrTx-rxϵ^2whenF′′rxϵ=0,where N(x)=-F′(r(xϵ^))(r(T(x))-r(xϵ^))+F′(r(xϵ))(λ(x)r(T(x))-r(xϵ^)) and Q(r(xϵ))=supF′′t for t∈(r(xϵ)-1,r(xϵ)+1). Note that we choose xϵ^ as close to xϵ such that sign[F′′(r(xϵ^))]=sign[F′′(r(xϵ))]. Therefore, H(x)-N(x)≥0.Letv(x)=F(r∘T(x))+F(r(xϵ))-F(r(xϵ^))+H(x). Then one gets v(xϵ)=F(r(xϵ))=u(xϵ). Because of the fact F′(r(x))∇r(x)=∇u(x)=G(r(x)2)-12r(x)∇r(x) and the inequality (41), we get(52)0<F′rx=Grx2-12rx<C2Grx-1.Moreover, we have two inequalities; that is, for x∈U,(53)first  order  term  ofvx-ux=F′rxϵ·λxrTx-rxϵ^-rx-rxϵ≥0,second  order  term  ofvx-ux=Hx-Nx≥0.Hence v is an upper-supporting function for u at the point xϵ.Since∇H|xϵ=∇N|xϵ, ∇λ|xϵ=0, λ(xϵ)=1, and ∇(r∘T)=1, we have (54)∇vxϵ≤F′rxϵ·∇λxϵrxϵ^+λxϵ∇r∘Txϵ=F′rxϵ=∇uxϵ<C2.By our assumption (2), the above inequality implies that(55)∇~xϵ1vxϵ<e1C2.Notice that(56)Δ~xϵr∘Txxϵ=DT2Δ~xϵrxϵ^=nΔ~xϵ^rxϵ^,where dim⁡M=n. By a simple calculation, we have (57)F′′rx∇rx=2Grx2-1-2rx2Grx2-1+1∇rxand hence(58)F′′rx=2Grx2-1-2rx2Grx2-1+1<2Grx2-1.Using ∇(r∘T)=1, ∇~xϵ(r∘T)≤d1, (52), (56), and (58), we have(59)Δ~xϵvxϵ≤d12F′′rxϵ^+F′rxϵ^Δ~xϵr∘Txϵ+Δ~xϵHxϵ≤F′rxϵΔ~xϵr∘Txϵ+d12F′′rxϵ^+F′′rxϵifF′′rxϵ>0,F′rxϵΔ~xϵr∘TxϵifF′′rxϵ<0,F′rxϵΔ~xϵr∘Txϵ+d12F′′rxϵ^+QrxϵifF′′rxϵ=0,(60)<12CGrxϵ-1nΔ~xϵ^rxϵ^+4d12Grxϵ2-1.Let 2a be the distance to a closest cut point of o. Because the point xϵ is a cut point of o, by (41) and (42), we get(61)2aGrxϵ2-1≤rxϵGrxϵ2-1<C4Grxϵ-1≤C4,(62)Grxϵ2-1<C8a.By plugging (62) to (60), our assumption (10) tells us that, for r>1,(63)Δ~xϵvxϵ<C2n+C2ad12.Therefore, by (55) and (63), we obtain, for r>1, (64)Lvxϵ<C2n+d12a+e1.So u satisfies the conditions for an L-tamed exhaustion function.Altogether, we can conclude thatu must be an L-tamed exhaustion function for M.



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