arXiv subjects
Publications and source records attributed to Kuntao Jin.
We prove that the product of three connected noncompact smooth manifolds without boundary admits a complete metric of uniformly positive scalar curvature. The proof uses the additivity of Morse indices on products and an orientation-free version of Das's open Morse--surgery construction. We also give an example of two open manifolds whose product admits no complete metric of nonnegative scalar curvature.
Let $(M^m,g)$, $m\geq2$, be a closed connected Riemannian manifold with $\secg_g\leq-1$, and let $(X,g_X)$ be its universal cover. Yau proved that $\hiso(X)\geq m-1$. We prove that equality is rigid: if $\hiso(X)=m-1$, then $(X,g_X)$ is isometric to $\Hh^m(-1)$.
Let $(M^3, g_{\mathrm{hyp}})$ be a closed oriented hyperbolic three-manifold normalized so that $\operatorname{sec}_{g_{\mathrm{hyp}}} \equiv -1$. We prove a sharp lower bound for the volume of hypersurfaces in $M^3 \times \mathbb{S}^1$ representing the slice class $[M^3 \times \{ \mathrm{pt} \}] \in H_3(M^3 \times \mathbb{S}^1; \mathbb{Z})$, and we classify the equality case. If $g$ is a smooth Riemannian metric on $M^3 \times \mathbb{S}^1$ with the scalar curvature $\operatorname{Sc}_g \geq -6$, then every closed embedded hypersurface $Σ$ representing the slice class $[M^3\times\{\mathrm{pt}\}]$ satisfies $\operatorname{vol}_g(Σ) \geq \operatorname{vol}_{g_{\mathrm{hyp}}}(M^3)$. The bound is attained by the product metric $g_{\mathrm{hyp}}+h$, with $h$ any metric on $\mathbb{S}^1$. Conversely, if equality holds for some $Σ$, then up to a diffeomorphism preserving the slice class, $g=g_{\mathrm{hyp}}+h$ and $Σ= M^3 \times \{\mathrm{pt}\}$.
Let \((M^m,g)\) be a closed Riemannian manifold with \(\sec_g\leq-1\). We prove that the bottom spectrum of its universal cover attains McKean's lower bound if and only if the universal cover is hyperbolic space of constant sectional curvature \(-1\). More generally, for every \(1<p<\infty\), the variational \(p\)-fundamental tone satisfies \[ λ_{1,p}(\wti M) \geq\left(\frac{m-1}{p}\right)^p, \] and equality for some \(p\in(1,\infty)\) holds if and only if \(\wti M\cong\bH^m(-1)\). In that case, equality holds for every \(p\in(1,\infty)\). The proof converts the two McKean defects of a minimizing sequence into a stationary probability measure on the compact horospherical suspension; heat-kernel positivity then forces its zero-defect support to contain a complete leaf.