Searcharxiv⌕ Search

arXiv subjects

Jia-Yong Wu

Publications and source records attributed to Jia-Yong Wu.

At least 19 recordsLinked to original sources
arXiv2026

Spectral comparison and splitting theorems for the infinity-Bakry-Emery Ricci curvature

In this paper, we prove the diameter comparison, the global weighted volume comparison and the splitting theorem in weighted manifolds when the infinity-Bakry-Emery Ricci curvature has a lower bound in the spectrum sense. Our results extend Antonelli-Xu's spectral Bonnet-Myers and Bishop-Gromov theorems, and Antonelli-Pozzetta-Xu's spectral splitting theorem to weighted manifolds. Our results are also some supplements of Chu-Hao's spectral diameter and global volume comparisons, and Yeung's spectral splitting theorem in weighted manifolds.

math.DG↗
arXiv2025

Willmore-type inequalities for closed hypersurfaces in weighted manifolds

In this paper, we prove some Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-Émery Ricci curvature. In particular, we give a sharp Willmore type inequality in steady gradient Ricci solitons. We also prove a sharp Willmore-like inequality in shrinking gradient Ricci solitons. Moreover, we characterize the equality cases of Willmore-type inequalities. These results can be regarded as weighted versions of Agostiniani-Fogagnolo-Mazzieri's Willmore-type inequality. As applications, we derive some sharp isoperimetric type inequalities in weighted manifolds under the existence assumption of a critical set of weighted isoperimetric functional.

math.DG↗
arXiv2025

Geometric inequalities and rigidity of gradient shrinking Ricci solitons

In this paper we prove that the Sobolev inequality, the logarithmic Sobolev inequality, the Schrödinger heat kernel upper bound, the Faber-Krahn inequality, the Nash inequality and the Rozenblum-Cwikel-Lieb inequality all equivalently exist on complete gradient shrinking Ricci solitons. We also obtain some integral gap theorems for compact shrinking Ricci solitons.

math.DG↗
arXiv2024

Diameter estimates for submanifolds in manifolds with nonnegative curvature

Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On the other hand, for a compact convex surface with boundary smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we can estimate its intrinsic diameter in terms of its mean curvature field integral and the length of its boundary. These results are supplements of previous work of Topping, Wu-Zheng and Paeng.

math.DG↗
arXiv2023

Shrinkers with curvature pinching conditions are compact

In this paper, we give various curvature pinching conditions such that shrinkers are compact. On one hand, we prove that shrinkers with positive Ricci curvature are compact when they have bounded curvature and certain curvature pinching conditions. On the other hand, we prove that shrinkers with certain asymptotically nonnegative sectional curvature are compact. As applications, some related classifications of shrinkers are provided.

math.DG↗
arXiv2022

Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons

On gradient shrinking Ricci solitons, we observe that the study of Schrödinger heat kernel seems to be more natural than the classical heat kernel. In this paper we derive sharp Gaussian upper bounds for the Schrödinger heat kernel on complete gradient shrinking Ricci solitons. As applications, we prove sharp upper bounds for the Green's function of the Schrödinger operator. We also prove sharp lower bounds for eigenvalues of the Schrödinger operator. These sharp cases are all achieved at Euclidean Gaussian shrinking Ricci solitons.

math.DG↗
arXiv2022

Time analyticity for heat equation on gradient shrinking Ricci solitons

On a complete non-compact gradient shrinking Ricci soliton, we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable. This growth condition is sharp. As an application, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with quadratic exponential growth on shrinkers.

math.DG↗
arXiv2022

Gap theorems for ends of smooth metric measure spaces

In this paper, we establish two gap theorems for ends of smooth metric measure space $(M^n, g,e^{-f}dv)$ with the Bakry-Émery Ricci tensor $\mathrm{Ric}_f\ge-(n-1)$ in a geodesic ball $B_o(R)$ with radius $R$ and center $o\in M^n$. When $\mathrm{Ric}_f\ge 0$ and $f$ has some degeneration outside $B_o(R)$, we show that there exists an $ε=ε(n,\sup_{B_o(1)}|f|)$ such that such a space has at most two ends if $R\leε$. When $\mathrm{Ric}_f\ge\frac 12$ and $f(x)\le\frac 14d^2(x,B_o(R))+c$ for some constant $c>0$ outside $B_o(R)$, we can also get the same gap conclusion.

math.DG↗
arXiv2022

Counting ends on shrinkers

In this paper we apply a geometric covering method to study the number of ends on shrinkers. On one hand, we prove that the number of ends on any complete non-compact shrinker is at most polynomial growth with fixed degree. On the other hand, we prove that any complete non-compact shrinker with certain volume comparison condition has finitely many ends. Some special cases of shrinkers are also discussed.

math.DG↗
arXiv2021

Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons

In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, we prove that the space of harmonic functions with fixed polynomial growth degree is finite dimensional. We also prove analogous results for ancient caloric functions. On the other hand, without any curvature condition, we prove sharp finite dimensional estimates for the space of Schrödinger functions with fixed polynomial growth degree.

math.DG↗
arXiv2021

Diameter estimate for closed manifolds with positive scalar curvature

For a simply connected closed Riemannian manifold with positive scalar curvature, we prove an upper diameter bound in terms of its scalar curvature integral, the Yamabe constant and the dimension of the manifold. When a manifold has a conformal immersion into a sphere, the dependency on the Yamabe constant is not necessary. The power of scalar curvature integral in these diameter estimates is sharp and it occurs at round spheres with canonical metric.

math.DG↗
arXiv2021

Gradient estimates for a nonlinear parabolic equation with Dirichlet boundary condition

In this paper, we prove Souplet-Zhang type gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with the compact boundary under the Dirichlet boundary condition when the Bakry-Emery Ricci tensor and the weighted mean curvature are both bounded below. As an application, we obtain a new Liouville type result for some space-time functions on such smooth metric measure spaces. These results generalize previous linear equations to a nonlinear case.

math.DG↗
arXiv2021

Gradient estimates for weighted harmonic function with Dirichlet boundary condition

We prove a Yau's type gradient estimate for positive $f$-harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces with compact boundary when the infinite dimensional Bakry-Emery Ricci tensor and the weighted mean curvature are bounded below. As an application, we give a Liouville type result for bounded $f$-harmonic functions with the Dirichlet boundary condition. Our results do not depend on any assumption on the potential function $f$.

math.DG↗
arXiv2021

Comparison geometry for integral radial Bakry-Émery Ricci tensor bounds

In this paper we prove mean curvature comparisons and volume comparisons on a smooth metric measure space when the integral radial Bakry-Émery Ricci tensor and the potential function or its gradient are bounded. As applications, we prove diameter estimates and eigenvalue estimates on smooth metric measure spaces. These results not only give a supplement of the author's previous results under integral Bakry-Émery Ricci tensor bounds, but also are generalizations of the Wei-Wylie's pointwise results.

math.DG↗
arXiv2021

Sharp gradient estimates on weighted manifolds with compact boundary

In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai.

math.DG↗
arXiv2021

Sharp upper diameter bounds for compact shrinking Ricci solitons

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman's entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.

math.DG↗
arXiv2018

Gradient estimates for a nonlinear parabolic equation and Liouville theorems

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positive smooth solutions to many special cases of the nonlinear equation. In particular, we apply gradient estimates to discuss some Yamabe-type problems of complete Riemannian manifolds and smooth metric measure spaces.

math.DG↗