arXiv research

We consider a class of complete Kahler manifolds with a strictly pseudoconvex boundary at infinity. After studying its asymptotic geometry, we formulate a conjecture in the Kahler-Einstein case relating the bottom of spectrum to the CR geometry on the boundary. We prove some partial results.

Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.

In \cite{LiWang2001complete1,LiWang2001complete2}, Li-Wang proved a splitting theorem for an n-dimensional Riemannian manifold with $Ric\geqslant -(n-1)$ and the bottom of spectrum $λ_0(M)=\frac{(n-1)^2}{4}$. For an n-dimensional compact manifold $M$ with $Ric\geqslant-(n-1)$ with the volume entropy $h(M)=n-1$, Ledrapp…

Survey on bottom of spectrum of Hodge Laplacian on complete noncompact Kähler manifolds

The study of spectral-tightness in Riemannian manifolds and its topological implications.

We prove a sharp integral gradient estimate for harmonic functions on noncompact Kähler manifolds. As application, we obtain a sharp estimate for the bottom of spectrum of the p-Laplacian and prove a splitting theorem for manifolds achieving this estimate.

Consider a compact Kähler manifold $M^m$ with Ricci curvature lower bound $Ric_M\geq -2(m+1) .$ Assume that its universal cover $$ has maximal bottom of spectrum $λ_1(\widetilde{M}$ Then we prove that $\widetilde{M}$ is isometric to the complex hyperbolic space $\Bbb{CH}^m.$

Let $(M^{n},g)$ be a compact Riemannian manifold with $Ric\geq-(n-1)$. It is well known that the bottom of spectrum $λ_{0}$ of its unverversal covering satisfies $λ_{0}\leq(n-1) ^{2}/4$. We prove that equality holds iff $M$ is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy.

Global solutions found for certain reaction-diffusion equations on specific manifolds.