Some Geometry And Analysis On Smooth Metric Measure Space

We divide it into four parts. The first part is Obata theorem and its generalizations, the second part is Self-Shrinker in Warped product space and weighted Minkowski inequality, the third part is Monotonicity for Bakry-Emery Ricci curvature, and the fourth part is gradient Ricci soliton.I. Obata theorem and its generalizationsWe consider the generalized Obata equation: If we assume w has at least one critical point, we have the following theorem.Theorem0.1. Let (M, g) be a connected complete Riemannian manifold of dimension n≥2which admits a nonconstant smooth solution w of the generalized Obata equa-tion (10) for a smooth function f(s). Assume that w has at least one critical point p. Then M is diffeomorphic to Rn or Sn. Moreover,(M, g) is isometric to Mf,μ with μ=w(p).If we give some conditions on f, it is not necessary to assume that w has a critical point.Theorem0.2. Let (M, g) be a connected complete Riemannian manifold of dimension n≥2which admits a nonconstant smooth solution of the generalized Obata equation (10) for a coercive function f. Then M=Mf,μ for some μ. In particular, M is diffeomorphic to Sn or Rn.Theorem0.3. Let (M, g) be a connected complete Riemannian manifold of dimension n which admits a nonconstant smooth solution of the generalized Obata equation (10) for a degenerately coercive function f. Then M is diffeomorphic to Rn. Moreover, if n≥2, then (M, g) is isometric to Mf,μ for some μ.Theorem0.4. Let (M, g) be a connected complete Riemannian manifold of dimension n which admits a nonconstant smooth solution of the generalized Obata equation (10) for a nondegenrately coercive function f. Then M is diffeomorphic to Sn. Moreover, ifn≥2, then (M, g) is isometric to Mf,μ for some μ.The following theorem implies that manifolds admit equation (1) have a warped product structure.Theorem0.5. Let (M, g) be a connected complete Riemannian manifold which admits a nonconstant smooth solution w of the generalized Obata equation (10) for some smooth f. Let I denote the interior of the image Iw ofw. Let μ∈I. Set N=ω-1(μ) and Ω=w-1(I). Then (N,gN) is connected and complete with the induced metric gN and there is a diffeomorphism F:I×N→Ω such that w(F(s,p))=s for all (s,p). The pullback metric F*g is a warped product metric given by the formula (2.40). Furthermore, there holds M=Ω and(?)Ω consists of at most two points. Each point is either a unique global maximum point or a unique global minimum point ofw.Conversely, if (N, gN) is a Riemannian manifold and h(s) a positive smooth func-tion on an interval I, then the function w=s on I x N satisfies the generalized Obata equation with f=-1/2h, where I x N is equipped with the metric h-1ds2+h/(α2)gN.Next we consider the hyperbolic case of generalized Obata equation. Let Wh(M, g) denotes the space of smooth solutions of equation (11).Theorem0.6. Let (M, g) be a connected complete Riemannian manifold of dimension n≥2. Set Wh=Wh(M,g). Then dim Wh≥n iff(M,g) is isometric to Hn. Consequently, if dim Wh≥n, then dim Wh=n+1.Theorem0.7. Let (M, g) be a connected complete Riemannian manifold of dimension n≥2. Set Wh=Wh(M, g). Then dim Wh=n-iff(M, g) has constant sectional curvature-1and is diffeomorphic to Rn-1x S1(equivalently, π1(M)=Z). More pre-cisely, dim Wh=n-1iff(M, g) is isometric to Hcosh(n-1)(S1(p)) or Hcosh(n-2)(Hexp(S1(p))) for some p>0.(The former contains a closed geodesic while the latter doesn’t.)The following theorem characterizes lower dimensions of Wh(M, g). Theorem0.8. Let (M, g) be a connected complete Riemannian manifold of dimension n and1≤k≤n-1. Then dimWh(M,g)≥kiff M is isometric to Hcoshk(N,gN) or Hcosh(k-1)(Hexp(N, gN)) for a connected complete Riemannian manifold (N, gN) of di-mension n-k.If (M, g) is non-complete, we still have similar theorems.Lemma0.1. If M is an open region of Sn, then dimWs(M,g)=n+1. Where Ws(M, g) denotes the space of smooth solutions of Obata equation▽dw+wg=0.Theorem0.9. Let (M, g) be a connected Riemannian manifold of dimension n. Then dim Ws(M, g)≥n iff(M, g) is isometric to a region of Sn.Lemma0.2. Wh(Hn,g)=span{x1,x2,…,xn,xn+1}.Theorem0.10. Let (M, g) be a connected Riemannian manifold of dimension n. Then dim Wh(M, g)≥n iff(M, g) is isometric to a region of Hn.Ⅱ. Self-Shrinker in Warped product space and weighted Minkowski inequalityAssume E is a closed, embedded, oriented hypersurface in Rn. Self-Shrinker equation is the following one, where ν is the outer normal vector of∑, H is the mean curvature and X is position vector.If we write the Euclidean metric as thenHere we consider the manifold M=N x [0, a) equipped with the Riemannian metric We also assume E is a closed, embedded, oriented hypersurface in M. Similar to (12), formally we define the self-shrinker equation as the outer normal vector of E and H is the mean curvature.The following is our result about the self-shrinker. Theorem0.11. If∑is a smooth hypersurface in M with positive mean curvature and satisfies H=(X,ν). If the ambient space M has nonnegative sectional curvature and R(ei, ej, ek, ν)=0, where ei, ej, ek are tangent vectors and ν is the outer normal vector of∑, then VA=0, i.e. the second fundamental form of∑is parallel.we consider the warped product space (M, g) equipped with the weighted measure dvol=e-fdvol(-g), where g is given as (3.3). Here we choose special f such that▽2f=φ’g, i.e. f(r)=∫0rφ(s)ds.We assume φ satisfies the following conditions.(H1)φ(r)=r·ξ(r2), whereξ:[0, α)→R is a smooth positive function satisfying ξ(0)＝1.(H2) φ’(r)>0for all r∈(0, α).(H3) The funcion is non-decreasing for r∈[0,α).(H4) φ’(r)2≥φ(r)φ"(r).We get the weighted Minkowski inequality.Theorem0.12. Let M be a warped product manifold satisfying (H1)-(H4), and∑is a closed hypersurface which is star shaped and Hf>0. Then the following inequality holds where Ω is the domain enclosed by∑.Ⅲ. Monotonicity for Bakry-Emery Ricci curvatureWith respect to the measure e-f dvol the natural self-adjoint f-Laplacian is Δf=Δ-▽f·▽. Consider the positive Green’s function G(x0,·) of the f-Laplacian of (Mn, g, e-f dvol)(see Definition4.6). For any real number k>2, let b=G1/(2-k). For β,ι,p∈R, when b is proper, we consider While Afβ(r) is well defined for all r>0, Vfβ,p(r) is only well defined whenSee the proof of Lemma4.9for detail. When k=ι=n,P=2,p=0, these reduce to A(r),V(r) in[23]. Whenk=ι=n,p=2, these are Aβ,Vβ in[26].First we obtain the following gradient estimate for b.Proposition0.1. If a smooth metric measure space (Mn, g, e-fdvol)(n>3) has RicfN>0, then for k=n+N, there exists r0>0, such that on M\B(x0, r0),If RicfN≥0, we get a monotonicity formula for some combination of A and V.Theorem0.13. IfMn(n>3) has (15) Hence if in addition β≥2, then Afβ-αVfβ,p is nondecreasing in r. If RicfN>0, we get the monotonicity for A.Theorem0.14. If Mn(n≥3) has RicfN>0, then forIf Ricf≥0, we also get the monotonicity for A and the linear combination of A and V. Theorem0.15. If Mn(n≥4)hasRicf≥0, then fo we have(we have (Ⅳ. Gradient Ricci solitonFirst we quote an useful lemma which is used in the following argument.Corollary0.1. Suppose that (M, g(t)) is an ancient Ricci flow such that for each t≤0the Riemannian manifold (M, g(t)) is complete and has nonnegative sectional curva-ture. Let V(t) be the asymptotic volume of the manifold (M,g(t)), then V(t) is α non-decreasing function of t.The following is the curvature bound of Ricci flow satisfying some conditions. Lemma0.3. Let (M,g(t)) be an immortal solution of the Ricci flow with Kg(t)C>0. Suppose that (M, g(0)) has Euclidean volume growth, then there is a uniform constant C, such that for any (p,t)∈M x (0,∞).As an application, we get the curvature bound for some gradient expanding soli-ton.Corollary0.2. Let (M, g) be a gradient expanding soliton with Kg(t)C≥0. Suppose (M, g) has Euclidean volume growth, then the curvature is bounded.