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Anqiang Zhu

Publications and source records attributed to Anqiang Zhu.

At least 19 recordsLinked to original sources
arXiv2026

The Faber-Krahn inequality for $p$-Hermite operators

We prove a Faber-Krahn inequality for the first eigenvalue of the $p$-Hermite operator (the weighted $p$-Laplacian with Gaussian weight) on Lipschitz domains in $\R^n$ under Robin boundary conditions with positive Robin parameter. The main result states that, among all domains of given Gaussian measure, the first eigenvalue is minimized by a half-space, and equality holds only for half-spaces. This extends the classical Faber-Krahn inequalities for the $p$-Laplacian \cite{BucurCV} to the $p$-Hermite operator and generalizes the linear case \cite{ChiacchioMathann} to the full nonlinear regime $p>1$.

math.SP↗
arXiv2026

An isoperimetric inequality for Neumann eigenvalues with radial log-concave measures

We prove a sharp isoperimetric inequality for the harmonic mean of the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, endowed with radial log-concave measures. The main novelty is that we establish the sharp harmonic mean inequality under general radial log-concave measures, without assuming the weight function to be non-increasing. This extends previous results that were restricted to specific or more restrictive weighted settings. The proof relies on a refined analysis of the first eigenfunction on geodesic balls, a monotonicity property derived from a convexity condition on the radial weight, and a matrix trace inequality.

math.SP↗
arXiv2026

Faber Krahn inequality of Robin eigenvalue of the Weighted Laplacian

In this paper, we study the isoperimetric inequalities for the Robin eigenvalues of the weighted Laplacian with positive Robin parameter in the Euclidean space $\R^n$ and the hyperbolic space $\mathbb{H}^n$, respectively. More precisely, we prove that among all bounded Lipschitz domains with fixed weighted volume, the geodesic ball centered at the origin minimizes the first Robin eigenvalue of the weighted Laplacian, provided that the Robin parameter and the radial log-convex density satisfy suitable conditions. Furthermore, we show that the second Robin eigenvalue is bounded below by the first Robin eigenvalue of the geodesic ball centered at the origin with half the weighted volume. Our results extend classical Faber-Krahn inequalities to the setting of weighted spaces with log-convex densities. We also derive a lower bound for the second Robin eigenvalue in terms of the first eigenvalue of the centered ball with half the weighted volume.

math.SP↗
arXiv2026

Upper Bounds for Hessian Matrices of Positive Solutions to Heat Equations on Kähler Manifolds

We prove global and local upper bounds for the Hessian matrices of positive solutions to the heat equation on Kähler manifolds whose bisectional curvature is bounded from below. We also improve a result of Han and Zhang by weakening the curvature assumptions in their Hessian estimates on Riemannian manifolds. More precisely, we extend their global and local upper bounds, originally obtained under two-sided curvature bounds, to Riemannian manifolds with sectional curvature bounded from below.

math.DG↗
arXiv2026

A note on the first variation of the total mass

In this paper, we establish a proof for the first variation formula of the total mass within the $L_p$ framework. Our main result removes an extra restrictive determinant condition imposed in a theorem originally proved by Fang,Xing and Ye

math.CA↗
arXiv2026

Comparison Results for a class of Neumann Problems of the $p$-Laplace Equation on Riemannian Manifolds

We consider Neumann boundary value problems for the $p$-Laplace equation on Riemannian manifolds with nonnegative Ricci curvature. Using spherical symmetrization under appropriate constraints, we derive Talenti-type comparison results in Lorentz spaces. We further show that, in contrast to the Robin case, the Neumann setting admits weaker constraints, which yields stronger comparison principles.

math.AP↗
arXiv2026

An isoperimetric inequality for the second Robin eigenvalue of the Weighted Laplacian

In this paper, we investigate a shape optimization problem for the second Robin eigenvalue of the weighted Laplacian on bounded Lipschitz domains symmetric about the origin. Our main theorem states that the ball centered at the origin maximizes the second Robin eigenvalue among all Lipschitz bounded domains of prescribed weighted measure and symmetric about the origin for a range of negative Robin parameters.

math.AP↗
arXiv2023

The spectral rigidity of Ricci soliton and Einstein-type manifolds

We are concerned in this article with a classical topic in spectral geometry dating back to McKean-Singer, Patodi and Tanno: whether or not the constancy of sectional curvature (resp. holomorphic sectional curvature) of a compact Riemannian manifold (resp. Kähler manifold) can be completely determined by the eigenvalues of its $p$-Laplacian for a \emph{single} integer $p$? We treat this question under two conditions: gradient shrinking Ricci soliton for Riemannian manifolds and cohomologically Einstein for Kähler manifolds. We show that, with some sporadic unknown cases, this is true for each $p$. Furthermore, we show that the condition of being isospectral can be relaxed to a suitable almost-isospectral version.

math.DG↗
arXiv2018

On the $κ-$solutions of the Ricci flow on noncompact 3-manifolds

In this paper we prove that there is no $κ$-solution of Ricci flow on 3-dimensional noncompact manifold with strictly positive sectional curvature and blow up at some finite time $T$ satisfying $\int^T_0 \sqrt{T-t} R(p_0,t)dt< \infty$ for some point $p_0$. This partially confirms a conjecture of Perelman.

math.DG↗
arXiv2015

Liouville theorem for warped ancient Ricci solutions

In this note, we answer affirmatively the question if a warped product of a compact manifold with a line as an ancient solution to the Ricci flow is trivial. We also consider the global behavior of the Type III warping product Ricci flow.

math.DG↗
arXiv2012

Yamabe flow and ADM Mass on asymptotically flat manifolds

In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert functional under the Yamabe flow. Through studying the Yamabe flow by weighted spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined and monotone non-increasing under the Yamabe flow on $n$-dimensional, $n\geq 3$, asymptotically flat manifolds. In the case of dimension $n=3$ or 4, we obtain that the ADM mass is invariant under the Yamabe flow and the Yamabe flow is the gradient flow of Einstein-Hilbert functional on asymptotically flat manifolds

math.DG↗
arXiv2012

On the Perelman's reduced entropy and Ricci flat manifolds with maximal volume growth

In this paper, we study the Ricci flat manifolds with maximal volume growth using Perelman's reduced volume of Ricci flow. We show that if $(M^n,g)$ is an noncompact complete Ricci flat manifold with maximal volume growth satisfying $|Rm|(x)\to 0$ as $d(x)=d_g(x,p)\to \infty$, then $M^n$ has the quadratic curvature decay. Some applications to this result are also presented.

math.DG↗
arXiv2011

On the weighted forward reduced Entropy of Ricci flow

In this paper, we first introduce the weighted forward reduced volume of Ricci flow. The weighted forward reduced volume, which related to expanders of Ricci flow, is well-defined on noncompact manifolds and monotone non-increasing under Ricci flow. Moreover, we show that, just the same as the Perelman's reduced volume, the weighted reduced volume entropy has the value $(4π)^{\frac{n}{2}}$ if and only if the Ricci flow is the trivial flow on flat Euclidean space.

math.DG↗
arXiv2008

On a length preserving curve flow

In this paper, we consider a new length preserving curve flow for convex curves in the plane. We show that the global flow exists, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C-infinity topology as t goes to infinity.

math.DG↗
arXiv2008

Nonsingular Ricci flow on a noncompact manifold in dimension three

We consider the Ricci flow $\frac{\partial}{\partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rm\geq 0, |Rm(p)|\to 0, ~as ~d(o,p)\to 0.$ We prove that the Ricci flow on such a manifold is nonsingular in any finite time.

math.DG↗