Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds

A gradient-entropy inequality is established for elliptic diffusion semigroups on arbitrary complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived.     

A Liouville-type result for quasi-linear elliptic equations on complete Riemannian manifolds

We extend a Liouville-type result of D. G. Aronson and H. F. Weinberger and E.N. Dancer and Y. Du concerning solutions to the equation Δ_pu=b(x)f(u) to the case of a class of singular elliptic operators on Riemannian manifolds, which include the ?-Laplacian and are the natural generalization to manifolds of the operators studied by J. Serrin and collaborators in Euclidean setting. In the process, we obtain an a priori lower bound for positive solutions of the equation in consideration, which complements an upper bound previously obtained by the authors in the same context.     

Prescribed Q-curvature problem on closed 4-Riemannian manifolds in the null case

The main objective of this short note is to give a sufficient condition for a non constant function k to be Q curvature candidate for a conformal metric on a closed Riemannian manifold with the null Q-curvature. In contrast to the prescribed scalar curvature on the two-dimensional flat tori, the condition we provided is not necessary as some examples show. The second author would like to thank Department of Mathematics, University of Paris XII for their invitation, hospitality and financial support during his visit. His research is partially supported by NUS research grant: R-146-000-077-112.     

On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds

We study the second best constant problem for logarithmic Sobolev inequalities on complete Riemannian manifolds and investigate its relationship with optimal heat kernel bounds and the existence of extremal functions.     

A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric

We first prove stochastic representation formulae for space–time harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems under appropriate curvature conditions. Space–time harmonic mappings which are defined globally in time correspond to ancient solutions to the harmonic map heat flow. As corollaries, we establish triviality of such ancient solutions in a variety of different situations.     

The global geometry of Riemannian manifolds with commuting curvature operators

We give manifolds whose Riemann curvature operators commute, i.e. which satisfy for all tangent vectors x_i in both the Riemannian and the higher signature settings. These manifolds have global geometric phenomena which are quite different for higher signature manifolds than they are for Riemannian manifolds. Our focus is on global properties; questions of geodesic completeness and the behaviour of the exponential map are investigated. Dedicated to the memory of Jean Leray     

A uniform bound for the solutions to a simple nonlinear equation on Riemannian manifolds

Let M be a complete noncompact manifold with Ricci curvature bounded below. In this note, we derive a uniform bound for the solutions to the nonlinear equation

     

Bounds for the first Dirichlet eigenvalue attained at an infinite family of Riemannian manifolds

LetM be a compact Riemannian manifold with smooth boundary M. We get bounds for the first eigenvalue of the Dirichlet eigenvalue problem onM in terms of bounds of the sectional curvature ofM and the normal curvatures of M. We discuss the equality, which is attained precisely on certain model spaces defined by J. H. Eschenburg. We also get analog results for Kähler manifolds. We show how the same technique gives comparison theorems for the quotient volume(P)/volume(M),M being a compact Riemannian or Kähler manifold andP being a compact real hypersurface ofM.Work partially supported by a DGICYT Grant No. PB94-0972 and by the E.C. Contract CHRX-CT92-0050 GADGET II.     

Spectral gap on Riemannian path space over static and evolving manifolds

In this article, we continue the discussion of Fang–Wu (2015) to estimate the spectral gap of the Ornstein–Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along with explicit estimates we study the short-time asymptotics of the spectral gap. The results are then extended to the path space of Riemannian manifolds evolving under a geometric flow. Our paper is strongly motivated by Naber's recent work (2015) on characterizing bounded Ricci curvature through stochastic analysis on path space.     

Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(?K,m)$-condition, where $m\in [n,\infty )$ and $K\ge 0$ are two constants. Moreover, we introduce the W-entropy and prove the W-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the $CD(?K,m)$-condition and on compact manifolds equipped with $(?K,m)$-super Ricci flows.     

The logarithmic entropy formula for the linear heat equation on Riemannian manifolds

In this paper we introduce a new logarithmic entropy functional for the linear heat equation on complete Riemannian manifolds and prove that it is monotone decreasing on complete Riemannian manifolds with nonnegative Ricci curvature. Our results are simpler version, without Ricci flow, of R.-G. Ye’s recent result (arXiv:math.DG/0708.2008). As an application, we apply the monotonicity of the logarithmic entropy functional of heat kernels to characterize Euclidean space.     

Isolated singularities of solutions to the Yamabe equation

In this paper we study the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not necessarily conformally flat. We are able to prove, when the dimension is less than or equal to five, that any solution is asymptotic to a rotationally symmetric Fowler solution. We also prove refined asymptotics if deformed Fowler solutions are allowed in the expansion.     

Weak convergence of laws of stochastic processes on Riemannian manifolds

Weak convergence of the laws of discrete time re-metrized stochastic processes derived from Brownian motions on compact Riemannian manifolds with heat kernels uniformly bounded by a constant on each compact set of the time parameter and bounded volumes to a stochastic process is given. With a weak condition, we also give weak convergence of those of Brownian motions themselves on manifolds in the same class. Several examples are given, which cover the cases when the manifolds collapse, the cases when the original Brownian motions converge to a non-local Markov process, and the cases when the Gromov-Hausdorff limit and the spectral limit by Kasue and Kumura are different. Received: 22 February 2000?Published online: 9 March 2001     

Schouten curvature functions on locally conformally flat Riemannian manifolds

Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor A _g associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σ _k (A _g ), 1 ≤ k ≤ n} of the eigenvalues of A _g with respect to g; we call σ _k (A _g ) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: A _g is semi-positive definite and σ _k (A _g ) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ₂(A _g ) is a non-negative constant and (M ⁿ , g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant scalar curvature and non-negative Ricci curvature. Udo Simon: Partially supported by Chinese-German cooperation projects, DFG PI 158/4-4 and PI 158/4-5, and NSFC.     

Log-Sobolev inequality on non-convex Riemannian manifolds

Counterexamples are constructed to show that when the second fundamental form of the boundary is bounded below by a negative constant, any curvature lower bound is not enough to imply the log-Sobolev inequality. This indicates that in the study of functional inequalities on non-convex manifolds, the concavity of the boundary cannot be compensated by the positivity of the curvature. Next, when the boundary is merely concave on a bounded domain, a criterion on the log-Sobolev inequality known for convex manifolds is proved. Finally, when the concave part of the boundary is unbounded, a Sobolev inequality for a weighted volume measure is established, which implies an explicit sufficient condition for the log-Sobolev inequality to hold on non-convex manifolds.     

Weitzenböck formulas for Riemannian foliations

In this paper we study the interplay between adiabatic limits of a Riemannian foliation and the classical Weitzenböck formula. For the leafwise part, our study leads to a vanishing result for the first order term of differential spectral sequence associated with the foliation. For the transversal part we obtain a Weitzenböck type formula which is an extension of the previous formula for basic forms due to Ph. Tondeur, M. Min-Oo, and E. Ruh, and is also more general than a Weitzenböck formula for transverse fiber bundle due to Y. Kordyukov.     

The L p spectrum of Riemannian products

We prove that the L ^p spectrum of a Riemannian product M ₁×M ₂ coincides with the set theoretic sum of the L ^p spectra of M ₁ and M ₂ . Received: 13 June 2007     

Minimizing weak solutions for calabi’s extremal metrics on toric manifolds

In this paper, we discuss Donaldson’s version of the modified K-energy associated to the Calabi’s extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified K-energy. The second author was partially supported by NSF10425102 in China and the Huo Y-T Fund.     

Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel

In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet p  -Laplacian (1<p<∞$1<p<\infty$) obtained by Matei (2000) [19] and Takeuchi (1998) [22], respectively. Moreover, we use this generalized eigenvalue comparison theorem to get estimates for the first eigenvalue of the Dirichlet p-Laplacian of geodesic balls on complete Riemannian manifolds with radial Ricci curvature bounded from below w.r.t. some point. In the rest of this paper, we derive an upper and lower bound for the heat kernel of geodesic balls of complete manifolds with specified curvature constraints, which can supply new ways to prove the most part of two generalized eigenvalue comparison results given by Freitas, Mao and Salavessa (2013) [9].