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 L 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 (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=TrA∘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)supMTrA+supMV<∞.
Definition 1.
A smooth complete Riemannian manifold M 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 operator Δ, 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.
Let o∈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.
Let u be a real-valued continuous function on M and let a point p∈M.
A function u is called proper, if the set {p:u(p)≤r} is compact for every real number r.
A function v 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 function u:M→R is called a Δ-tamed exhaustion, if the following condition holds:
u≥0.
At all points p∈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.
Let o∈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.
Let o∈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 Laplacian Δ 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 Theorem 5
The proof is similar to the method in [5]. Let U=suph. 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 function F(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-suphx: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>hxif rx≤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=infλ: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.
Let λ0>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 that hλ0 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=rxϵfor i=1 or 2.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. Since γ 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 of F, 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 Theorem 6
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 function u 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)=logϕ(r(x)2). By a direct calculation, one gets(39)∇u=∇logϕr2=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~xϵlogϕr2=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+C2on M∖Bxϵ,r0.Thus, by (45) and (48), we have (49)Lu=Δ~xϵu+∇~xϵ1u<C2d12+1+e1on M∖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.
Let xϵ be a cut point of o and let F(t)=logϕ(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 for u.
For a neighborhood U⊂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^-rxϵon U.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^2when F′′rxϵ>0,Nx-12F′′rxϵ^rTx-rxϵ^2when F′′rxϵ<0,Nx+12QrxϵrTx-rxϵ^2when F′′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.
Let v(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 of vx-ux=F′rxϵ·λxrTx-rxϵ^-rx-rxϵ≥0,second order term of vx-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 dimM=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′′rxϵif F′′rxϵ>0,F′rxϵΔ~xϵr∘Txϵif F′′rxϵ<0,F′rxϵΔ~xϵr∘Txϵ+d12F′′rxϵ^+Qrxϵif F′′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 that u must be an L-tamed exhaustion function for M.
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].
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