(东瑜昕) (林和子) (陆琳根),
1. School of Mathematical Sciences, Fudan University, Shanghai 20043, China;
2. School of Mathematics and Statistics & Laboratory of Analytical Mathematics and Applications(Ministry of Education) & FJKLMAA, Fujian Normal University, Fuzhou 350108, China E-mail: yxdong@fudan.edu.cn; lhz1@fjnu.edu.cn; lulingen@fudan.edu.cn
Abstract In this note, we prove a logarithmic Sobolev inequality which holds for compact submanifolds without a boundary in manifolds with asymptotically nonnegative sectional curvature.Like the Michale-Simon Sobolev inequality, this inequality contains a term involving the mean curvature.
Key words asymptotically nonnegative sectional curvature; logarithmic Sobolev inequality; ABP method
The classical logarithmic Sobolev inequality,first proven by Gross[7],is a very useful tool in analysis and geometric evolution problems (cf.[4, 8, 11]).In 2000, Ecker [6] gave a logarithmic Sobolev inequality which holds for submanifolds in Euclidean space.In 2020, using the ABP technique, Brendle [2] established a sharp logarithmic Sobolev inequality for submanifolds in Euclidean space without a boundary.He[3]also gave several Sobolev inequalities for manifolds with nonnegative curvature by using the same technique.Combining the method in [3] with some comparison theorems, the authors of [5] proved some Sobolev inequalities for manifolds with asymptotically nonnegative curvature.In 2021, Yi and Zheng [10] proved a logarithmic Sobolev inequality for compact submanifolds without a boundary in manifolds with nonnegative sectional curvature.In this paper, we generalize the results of [2, 10] to the case where the ambient space has asymptotically nonnegative sectional curvature.This curvature notion was first introduced by Abresch[1].We will use some comparison results for these kinds of manifolds in order to prove our results.Complete manifolds with asymptotically nonnegative sectional curvature belong to the class of complete manifolds with radial sectional curvature bounded from below.Readers may find more general comparison results for manifolds with radial sectional curvature bounded below in [9, 12].
In this section, we follow closely the exposition of [5].Letλ(t) : [0,+∞)→[0,+∞) be a nonnegative and nonincreasing continuous function satisfing that
Recall that a complete noncompact Riemannian manifold (M,g) of dimensionn+pis said to have asymptotically nonnegative sectional curvature if there is a base pointo ∈Msuch that
is not increasing on [0,+∞), and thus we may introduce the asymptotic volume ratio ofMby
Obviously,P(0)=1, andP(t) is a nonnegative decreasing function.
By combining the ABP-method in [2, 3, 10] with some comparison theorems, we obtain a logarithmic Sobolev inequality which holds for submanifolds without a boundary in manifolds with asymptotically nonnegative sectional curvature as follows:
Theorem 1.1LetMbe a complete noncompact(n+p)-dimensional manifold of asymptotically nonnegative sectional curvature with respect to a base pointo ∈M.Let Σ be a compactn-dimensional submanifold ofMwithout a boundary, and letfbe a positive smooth function on Σ.Then
wherer0= max{d(o,x)|x ∈Σ},His the mean curvature vector of Σ,θis the asymptotic volume ratio ofMgiven by (1.5),b0andb1are defined as in (1.1) and (1.2).
whereθis the asymptotic volume ratio ofManddmax(x,Σ)=max{d(x,y)|y ∈Σ}.
ProofNoting thatr0=max{d(y,o)|y ∈Σ}, using the triangle inequality, we get that
This completes the proof.□
In this section, we assume that the ambient spaceMis a complete noncompact (n+p)-dimensional Riemannian manifold of asymptotically nonnegative sectional curvature with respect to a base pointo ∈M.Let Σ⊂Mbe a compact submanifold of dimensionnwithout a boundary, and letfbe a positive smooth function on Σ.Letdenote the Levi-Civita connection ofMand letDΣdenote the induced connection on Σ.The second fundamental formBof Σ is given by
for allx ∈Σ.Define the transport map Φr:T⊥Σ→Mby
This completes the proof of Theorem 1.1.□
Conflict of InterestThe authors declare no conflict of interest.