arXiv research

The paper shows inequality and rigidity for manifolds with integral Ricci curvature.

Willmore-type inequalities for bounded domains in manifolds with curvature bounds.

The paper examines the stability of Minkowski inequality for nearly spherical domains.

The paper establishes Harnack inequalities for solutions of nonlinear parabolic equations on manifolds with integral Ricci curvature bounds.

Study on surfaces in product space with curvature inequality.

We present in this note a lower bound for the Calabi functional in a given Kähler class. This yields an integral inequality for constant scalar curvature metrics, which can be viewed as a refined version of Yau's Chern number inequality.

Researchers prove a new inequality for special Riemannian manifolds.

We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, some rigidity and degeneracy theorems for holomorphic maps without assuming any pointwise curvature signs for both the domain and target manifolds are proved, in which key roles are played by total integration of the f…

The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.

The paper derives inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.

The paper proves geometric inequalities in sphere using locally constrained flows.

The study finds conditions for almost-Kähler 4-manifolds to be Kähler.

New proof shows certain 3D spaces are essentially like infinite space.

The paper proves inequalities under Bakry-Émery-Ricci curvature bounds.

A well known question in differential geometry is to control the constant in isoperimetric inequality by intrinsic curvature conditions. In dimension 2, the constant can be controlled by the integral of the positive part of the Gaussian curvature. In this paper, we showed that on simply connected conformal flat manifol…

The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.

The purpose is to study the CR-manifold with a contact structure conformal to the Heisenberg group. In our previous work \cite{WY}, we have proved that if the $Q'$-curvature is nonnegative, and the integral of $Q'$-curvature is below the dimensional bound $c_1'$, then we have the isoperimetric inequality. In this paper…

Two inequalities for convex surfaces in electrostatics.

The optimality of the integral inequality $\int\limits_γ\sqrt{k_1^2+k_2^2+k_3^2}ds>2π$ for closed curves with non-vanishing curvatures in $\mathbb R^4$ is discussed. We prove that an arbitrary closed curve of constant positive curvatures in $\mathbb R^4$ satisfies the inequality $\int\limits_γ\sqrt{k_1^2+k_2^2+k_3^2}ds…

In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow…

The paper studies curvature flows in hyperbolic space and proves geometric inequalities.

Study curvature invariants in sub-Riemannian manifolds.

We generalize an inequality of E. Heintze and H. Karcher [8] for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for $k$-Ricci curvature. Even in the case of a pointwise bound, this generalizes the classical inequality by replacing a sectional curvature bound with a $k$-Ricci b…

Derives inequalities for eigenvalues and renormalized volume of Poincaré-Einstein manifolds.

Study on gradient h-almost Yamabe solitons with scalar curvature estimation.

The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.

Paper compares total quotient curvature and proves bounds for Einstein metric.

In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space $\H^n$, by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov-Fenchel inequalities.

In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.

On a compact Riemannian manifold with boundary, we study how Ricci curvature of the interior affects the geometry of the boundary. First we establish integral inequalities for functions defined solely on the boundary and apply them to obtain geometric inequalities involving the total mean curvature. Then we discuss rel…

New method improves curvature estimates for stable surfaces.

We consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass as an integral of the product of the scalar curvature and a nonnegative potential function, thus provin…

A new method using mod n covering improves systolic inequalities.

Study extends compactness theorems to weighted manifolds with integral curvature bounds.

Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.

Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $Ω$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geome…

The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.

In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide rigidity results by the integral inequalities involving the Weyl curvature, the t…

We prove some isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We then relate them to inequalities involving the higher order mean-curvature integrals. We also apply our results to obtain sharp eigenvalue estimates and some sharp geometric inequalities i…

In the first part of this paper we prove some new Poincaré inequalities, with explicit constants, for domains of any hypersurface of a Riemannian manifold with sectional curvatures bounded from above. This inequalities involve the first and the second symmetric functions of the eigenvalues of the second fundamental for…

New weighted geometric inequalities for hypersurfaces in R^n proved.

The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.

New inequality controls domain volume for manifolds with large spectrum.

In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R^2. We derive a family of parametric inequalities involving the following geometric functionals associated to a given convex curve with a simple Fourier series proof: length, area of the region included by the …

We generalize Brendle's geometric inequality considered in \cite{B} to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperb…

Study p-Laplacian equation on Riemannian manifolds with positive Ricci curvature.

The paper proves rigidity for hypersurfaces with constant shifted curvature functions in warped product manifolds.

The study improves Bochner inequality on Finsler manifolds to derive important inequalities.