arXiv research

The paper proves inequalities for hypersurfaces in weighted manifolds.

New proof of Willmore inequality using geometric divergence inequality.

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

Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.

Paper extends Willmore inequality to manifolds with negative Ricci curvature.

Study on stability of free boundary Willmore problem using new gradient inequality.

The Willmore flow preserves surface volume, leading to convergence to a sphere.

Proves strict inequality for minimizers of Willmore energy under isoperimetric constraints.

The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.

The paper proves a Willmore-type inequality for unbounded convex sets.

In this article, we prove a geometric inequality for star-shaped and mean-convex hypersurfaces in hyperbolic space by inverse mean curvature flow. This inequality can be considered as a generalization of Willmore inequality for closed surface in hyperbolic $3$-space.

Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.

Li-Yau inequality applied to curves in 2D space.

Let $x:M\to S^{n+p}$ be an $n$-dimensional submanifold in an $(n+p)$-dimensional unit sphere $S^{n+p}$, $x:M\to S^{n+p}$ is called a Willmore submanifold to the following Willmore functional: $\int_M(S-nH^2)^{\frac{n}{2}}dv,$ where $S=\sum\limits_{α,i,j}(h^α_{ij})^2$ is the square of the length of the second fundam…

The biharmonic flow and Willmore flow are studied in higher dimensions using geometric evolution equations.

The study classifies surfaces in Berger spheres as Willmore and Hopf tori.

Global existence of Willmore flow with boundary via Li-Yau inequality.

Study examines surfaces with bounded fractional mean curvature, proving control over local parametrization.

In this paper we build an explicit example of a minimal bubble on a Willmore surface, showing there cannot be compactness for Willmore immersions of Willmore energy above $16 π$. Additionnally we prove an inequality on the second residue for limits sequences of Willmore immersions with simple minimal bubbles. Doing so,…

Paper proves anisotropic Minkowski inequality and related inequalities.

The paper proves inequalities for closed surfaces involving mean curvature.

Lower bounds on geodesic lengths for spheres with Willmore energy.

New proofs of unique photon surfaces in 4D spacetimes, extending previous work.

Sharp criteria for 2-varifolds to be induced by smooth immersions.

Inverse mean curvature flow converges to a disk in hyperbolic space.

Simon type monotonicity formulas for the Willmore functional $\int | \mathbf{H} |^2$ in the hyperbolic space $\mathbb{H}^n$ and $\mathbb{S}^n$ are obtained. The formula gives a lower bound of $\int_Σ | \mathbf{H} |^2$ where $Σ^2$ is any closed surface in $\mathbb{H}^n$.

Study of Willmore energy on sphere sublevel sets and flow singularities.

Willmore flow converges globally for surfaces with rotational symmetry below a specific energy threshold.

Stability of branched immersions with energy constraints.

We use the inverse mean curvature flow with a free boundary perpendicular to the sphere to prove a geometric inequality involving the Willmore energy for convex hypersurfaces of dimension $n\geq 3$ with boundary on the sphere.

In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in \cite{Luo} to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in $\mathbb{S}^5$, and then we use this relati…

Delaunay tori minimize Willmore energy under isoperimetric constraints.

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on t…

This is the second of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the Möbius degeneration of the tori. In the first paper the construction was perfo…

Paper proves a conjecture about minimal hypersurfaces in spheres.

Sharp inequality found for hypersurfaces in curved spaces.

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkow…

This paper explains a technique for proving geometric inequalities.

Study on bending energy of surfaces with curvature concentration, deriving new lower bounds.

In this article, by following the method in \cite{PT}, combining Willmore energy with isoperimetric inequalities, we construct two examples of singularities under mean curvature flow in $\mathbb{H}^3$. More precisely, there exists a torus, which must develop a singularity under MCF before the volume it encloses decreas…

Derives monotonic quantities for $p$-harmonic functions on manifolds.

In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial Ω$ in $M$, with equality holding true if and only if $(M{\setminus}Ω, g)$ is iso…

New comparison theorem for submanifolds with geometric inequalities.

The Willmore energy for Frenet curves in quaternionic projective space is the generalization of the Willmore functional for immersions into the 4-sphere. Critical points of the Willmore energy are called Willmore curves in quaternionic projective space. Using a Baecklund transformation on Willmore curves, we generalize…

The paper proves geometric inequalities for hypersurfaces in weighted manifolds.

The paper proves formulas for capillary surfaces and applies them to inequalities and area estimates.

Willmore flow preserves low energy surfaces to planes.

In this work, we study the Willmore submanifolds in a closed connected Riemannian manifold which are orbits for the isometric action of a compact connected Lie group. We call them homogeneous Willmore submanifolds or Willmore orbits. The criteria for these special Willmore submanifolds is much easier than the general t…