arXiv research

Study constant mean curvature surfaces with integrable boundary conditions.

The paper generalizes Alexandrov theorems for null hypersurfaces with integral curvature conditions.

We use an isomorphism between the space of valence two Killing tensors on an n-dimensional constant sectional curvature manifold and the irreducible GL(n+1)-representation space of algebraic curvature tensors in order to translate the Nijenhuis integrability conditions for a Killing tensor into purely algebraic integra…

We give various estimates of the first eigenvalue of the $p$-Laplace operator on closed Riemannian manifold with integral curvature conditions.

The paper improves curvature estimates for Ricci flow solutions with bounded scalar curvature.

We study conformally flat surfaces with prescribed Gaussian curvature, described by solutions $u$ of the PDE: $Δu(x)+K(x)\exp(2u(x))=0$, with $K(x)$ the Gauss curvature function at $x\in\RR^2$. We assume that the integral curvature is finite. For radially symmetric $K$ we introduce the notion of a least integrally curv…

Study of Killing spinor-valued forms and their integrability conditions.

We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature on a finite time interval $[0,T)$ can be extended over time $T$. Moreover, we show that the condition is optimal in some sense.

The study examines stability of metric measure spaces with integral Ricci curvature bounds.

Along a Ricci flow solution on a closed manifold, we show that if Ricci curvature is uniformly bounded from below, then a scalar curvature integral bound is enough to extend flow. Moreover, this integral bound condition is optimal in some sense.

We prove a sharp Zhong-Yang type eigenvalue lower bound for closed Riemannian manifolds with control on integral Ricci curvature.

The geometry of oscillatory integrals on manifolds with intermediate symmetry.

In this paper, we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a Riemannian manifold with codimension $d\geq1$, which generalizes the extension theorem for the mean curvature flow of hypersurfaces due to Le-Šešum \cite{LS} and the authors \cite{XYZ1,XYZ2}. Usin…

Study short-time existence of Ricci-DeTurck flow from rough metrics with Morrey-type integrability.

The paper estimates area and volume for spacetimes with integral mean curvature bounds.

Quantitative Sobolev extensions lead to Neumann heat kernel bounds.

The paper studies Ricci flow with finite curvature integrals on manifolds.

Improved bounds for Carleson-Sjölin operators on manifolds with specific curvature conditions.

Generalizes Ricci flow starting from small curvature concentration with a Morrey-type condition.

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

Study on scalar curvature deformations in pseudohermitian manifolds.

The paper derives new gradient estimates for nonlinear elliptic equations under integral Ricci curvature bounds.

Example surfaces with curvature bounds but no Laplacian eigenvalue lower bound.

Analyzes $L^{2}$-harmonic forms on curved manifolds, proving integrability conditions.

Characterizes a class of almost Hermitian 4-manifolds using integral identities.

We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the $L^2$ Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.

Paper bounds the lowest spectrum of manifolds with curvature constraints.

In this note we characterize compact hypersurfaces of dimension $n\geq 2$ with constant mean curvature $H$ immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when $n\geq 3$ and $H\neq 0$, they are locally contained in a rotational h…

In this note, we first prove that the solution of mean curvature flow on a finite time interval $[0,T)$ can be extended over time $T$ if the space-time integration of the norm of the second fundamental form is finite. Secondly, we prove that the solution of certain mean curvature flow on a finite time interval $[0,T)$ …

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

The article characterizes gradient ρ-Einstein solitons under specific conditions.

Upper bounds on Betti numbers via curvature constraints.

New connections found between curvature and Euler characteristic using Schrödinger operators.

The paper explores geometry and positive scalar curvature on non-compact manifolds.

We study closed $n$-dimensional manifolds of which the metrics are critical for quadratic curvature functionals involving the Ricci curvature, the scalar curvature and the Riemannian curvature tensor on the space of Riemannian metrics with unit volume. Under some additional integral conditions, we classify such manifol…

In this article, we introduce a $2$-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in \cite{LX} for substatic manifolds. On the other hand, this technique leads to various geometri…

Extends Calabi operator to Riemannian locally symmetric spaces.

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…

Revisits conformal metrics with finite Q-curvature, providing necessary and sufficient conditions.

The study classifies singularities in curved 3D shapes.

We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the $L^2$-norm of their scalar curvature and…

The paper proves conditions for Einstein solitons to split into line and manifold.

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-di…

In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvatur…

Let $M^n(n\geq3)$ be an $n$-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that $M^n$ satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds with harmonic curvature and positive scalar curvature. In particular, Theorem 1.4,…

Study on biharmonic hypersurfaces in spheres and space forms, proving rigidity under scalar curvature condition.

We classify, up to a local isometry, all non-Kahler almost Kahler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kahler 4-manifolds satisfy the third curvature condition of A. Gr…

New rigidity results for tensors on non-compact manifolds with curvature conditions.