arXiv subjects
Publications and source records attributed to Jing Mao.
In this paper, for complete manifolds with a modified Ricci curvature bounded from below, we can successfully set up Cheng-type eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and the weighted $p$-Laplacian on geodesic balls of these manifolds.
In this paper, we firstly establish weighted heat kernel comparison theorems for the weighted heat equation on complete manifolds with radial curvatures bounded, and then by mainly using this conclusion, we can obtain two eigenvalue comparison theorems for the first Dirichlet eigenvalue of the Witten-Laplacian as applications in spectral geometry. Besides, as byproducts, eigenvalue comparisons for the first Dirichlet eigenvalue of the weighted $p$-Laplacian ($1<p<\infty$) have been established as well.
In this paper, under suitable geometric constraints, we have successfully obtained characterizations for the extremum values of the functional of mixed eigenvalues of the Laplacian on triangles (or trapezoids) in the Euclidean plane $\mathbb{R}^2$.
In this paper, we establish a generalized sphere theorem for compact Riemannian manifolds with radial Ricci curvature bounded.
In this paper, we successfully establish a Courant-type nodal domain theorem for both the Dirichlet eigenvalue problem and the closed eigenvalue problem of the Witten-Laplacian. Moreover, we also characterize the properties of the nodal lines of the eigenfunctions of the Witten-Laplacian on smooth Riemannian $2$-manifolds. Besides, for a Riemann surface with genus $g$, an upper bound for the multiplicity of closed eigenvalues of the Witten-Laplacian can be provided.
For a given smooth convex cone in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$ which is centered at the origin, we investigate the evolution of strictly mean convex hypersurfaces, which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly, along an inverse curvature flow with the speed equal to $\left(f(r)H\right)^{-1}$, where $f$ is a positive function of the radial distance parameter $r$ and $H$ is the mean curvature of the evolving hypersurfaces. The evolution of those hypersurfaces inside the cone yields a fully nonlinear parabolic Neumann problem. Under suitable constraints on the first and the second derivatives of the radial function $f$, we can prove the long-time existence of this flow, and moreover the evolving hypersurfaces converge smoothly to a piece of the round sphere.
In this paper, by imposing suitable assumptions on the weighted function, (under the constraint of fixed weighted volume) a Brock-type isoperimetric inequality for Steklov-type eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven. This conclusion is actually an interesting extension of F. Brock's classical result about the isoperimetric inequality for Steklov eigenvalues of the Laplacian given in the influential paper [Z. Angew. Math. Mech. 81 (2001) 69-71]. Besides, a related open problem has also been proposed in this paper.
An isoperimetric inequality for lower order nonzero Neumann eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven in this paper. About this conclusion, we would like to point out two things: It strengthens the well-known Szeg\H{o}-Weinberger inequality for nonzero Neumann eigenvalues of the classical free membrane problem given in [J. Rational Mech. Anal. 3 (1954) 343-356] and [J. Rational Mech. Anal. 5 (1956) 633-636]; Recently, Xia-Wang [Math. Ann. 385 (2023) 863-879] gave a very important progress to the celebrated conjecture of M. S. Ashbaugh and R. D. Benguria proposed in [SIAM J. Math. Anal. 24 (1993) 557-570]. It is easy to see that our conclusion here covers Xia-Wang's this progress as a special case. In this paper, we have also proposed two open problems which can be seen as a generalization of Ashbaugh-Benguria's conjecture mentioned above.
In this paper, by mainly using the rearrangement technique and suitably constructing trial functions, under the constraint of fixed weighted volume, we can successfully obtain several isoperimetric inequalities for the first and the second Dirichlet eigenvalues, the first nonzero Neumann eigenvalue of the Witten-Laplacian on bounded domains in space forms. These spectral isoperimetric inequalities extend those classical ones (i.e. the Faber-Krahn inequality, the Hong-Krahn-Szeg\H{o} inequality and the Szeg\H{o}-Weinberger inequality) of the Laplacian.
Given an eigenvalue $\lambda$ of the Laplace-Beltrami operator on $n-$spheres or $-$hemispheres, with multiplicity $m$ such that $\lambda=\lambda_{k}=\dots = \lambda_{k+m-1}$, we characterise the lowest and highest orders in the set $\left\{k,\dots,k+m-1\right\}$ for which P\'{o}lya's conjecture holds and fails. In particular, we show that P\'{o}lya's conjecture holds for hemispheres in the Neumann case, but not in the Dirichlet case when $n$ is greater than two. We further derive P\'{o}lya-type inequalities by adding a correction term providing sharp lower and upper bounds for all eigenvalues. This allows us to measure the deviation from the leading term in the Weyl asymptotics for eigenvalues on spheres and hemispheres. As a direct consequence, we obtain similar results for domains which tile hemispheres. We also obtain direct and reversed Li-Yau inequalities for $\mathbb{S}^2$ and $\mathbb{S}^4$, respectively.
Let $Ω$ be a bounded domain (with smooth boundary) on the hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$. In this paper, by using a priori estimates, we can establish Pogorelov type estimates of $k$-convex solutions to a class of Hessian quotient equations defined over $Ω\subset\mathscr{H}^{n}(1)$ and with the vanishing Dirichlet boundary condition.
In this paper, under suitable settings, we can obtain the existence of solutions to a class of prescribed Weingarten curvature equations in warped product manifolds of special type by the standard degree theory based on the a priori estimates for the solutions. This is to say that the existence of closed hypersurface (which is graphic with respect to the base manifold and whose $k$-th Weingarten curvature satisfies some constraint) in a given warped product manifold of special type can be assured.
In this paper, under suitable settings, we can obtain the existence and uniqueness of solutions to a class of Hessian quotient equations with Dirichlet boundary condition in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, which can be seen as a prescribed curvature problem and a continuous work of [12].
In this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, and through which the existence of spacelike admissible graphic hypersurfaces, with prescribed $2$-th Weingarten curvature and Dirichlet boundary data, defined over a strictly convex domain in the hyperbolic plane $\mathscr{H}^{n}(1)\subset\mathbb{R}^{n+1}_{1}$ of center at origin and radius $1$, can be proven.
In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along the inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of hyperbolic plane of center at origin and prescribed radius, which actually corresponds to a constant function defined over the piece of $\mathscr{H}^{n}(1)$, as time tends to infinity.
In this paper, we consider the evolution of spacelike graphic curves defined over a piece of hyperbola $\mathscr{H}^{1}(1)$, of center at origin and radius $1$, in the $2$ dimensional Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$ along an anisotropic inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic curves converge smoothly to a piece of hyperbola of center at origin and prescribed radius, which actually corresponds to a constant function defined over the piece of $\mathscr{H}^{1}(1)$, as time tends to infinity.
In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along the inverse Gauss curvature flow (i.e., the evolving speed equals the $(-1/n)$-th power of the Gaussian curvature) with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of the spacelike graph of a positive constant function defined over the piece of $\mathscr{H}^{n}(1)$ as time tends to infinity.
In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or a Hadamard manifold, and of a biharmonic Steklov problem on bounded domains of a Euclidean space. Especially, interesting rigidity results can be obtained if sharp bounds were achieved.