arXiv research

Study of critical tori for mean curvature energies in Killing submersions.

New calibration energy measures deviation from calibrated geometry, enabling mean curvature flow in infinite volumes.

Study quantizes energy distribution in inhomogeneous phase transitions.

Proves existence of special 2-spheres in curved 3-spaces.

An energy estimate is proved for the Bel--Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an $L^{\infty}$ bound on the second fundamental form, and a bound on a spacetime version of Bel--Robinson energy.

The paper examines geometric curvatures in generalized Riemannian spaces.

Study minimizes Willmore energy with constraints on surface properties.

The study finds surfaces with constant anisotropic mean curvature foliated by circles in Euclidean space.

Study bounds CMC surface index in 3-manifolds using energy.

New theorem on 3-manifolds with curvature and convex boundary.

Study on surfaces minimizing elastic energy with boundary constraints.

Energy quantization for surfaces with area, volume, and mean curvature constraints.

Inspired by the work of Leung-Wan, we study the mean curvature flow in hyperkähler manifolds starting from hyper-Lagrangian submanifolds, a class of middle dimensional submanifolds, which contains the class of complex Lagrangian submanifolds. For each hyper-Lagrangian submanifold, we define a new energy concept called …

Study on surface configurations with curvature and elasticity.

Classifies soap film surfaces with vertical potentials.

Generic level sets in mean curvature flow are BV solutions.

In this paper, we prove the following theorem regarding the Wang-Yau quasi-local energy of a spacelike two-surface in a spacetime: Let $Σ$ be a boundary component of some compact, time-symmetric, spacelike hypersurface $Ω$ in a time-oriented spacetime $N$ satisfying the dominant energy condition. Suppose the induced me…

The paper proves inequalities for closed surfaces involving mean curvature.

We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes account of area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied…

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

The paper finds new constant $p$-mean curvature surfaces in the Heisenberg group.

The study proves a new positive energy theorem for manifolds with specific curvature properties.

Paper introduces a modified Allen-Cahn equation for better energy equipartition.

We first partially extend a theorem of Topping, on the relation between mean curvature and intrinsic diameter, from immersed submanifolds of $\mathbb{R} ^{n}$ to almost everywhere immersed, closed submanifolds of a compact Riemannian manifold. We use this to prove quantization of energy for pseudo-holomorphic closed c…

Proposes a new quasi-local mass for timelike 2-surfaces in spacetimes.

The paper establishes curvature inequalities and rigidity results for surfaces in Riemannian and Lorentzian geometry.

Study shows thresholding scheme converges for mean curvature flow of convex sets.

Study p-Willmore disks with boundary energies, finding equilibrium configurations.

In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqu…

The study shows that certain graphs are regular at boundary points.

Paper proves uniqueness of specific spacetime surfaces in a lightcone.

Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…

A neural flow method minimizes Willmore energy for 2-surfaces in 3D space.

Study null energy condition impacts on special hypersurfaces in static spacetimes.

The paper bounds the energy index of harmonic Gauss maps on surfaces.

Study of critical points for 4D conformally invariant curvature energies.

The paper proves rigidity estimates for varifolds almost minimizing the Willmore energy.

In this paper, we will derive a small energy regularity theorem for the mean curvature flow of arbitrary dimension and codimension. It says that if the parabolic integral of $|A|^2$ around a point in space-time is small, then the mean curvature flow cannot develop singularity at this point. As an application, we can pr…

We give an extensive treatment of the Constant Mean Curvature (CMC) Einstein flow from the point of view of the Bel-Robinson energies. The article, in particular, stresses on estimates showing how the Bel-Robinson energies and the volume of the evolving states control intrinsically the flow along evolution. The treatme…

Planes and spheres are the only stationary surfaces with constant Gauss curvature.

An anisotropic surface energy is the integral of an energy density that depends on the normal at each point over the considered surface, and it is a generalization of surface area. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wul…

In [8] Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Further…

The paper establishes curvature inequalities and rigidity results for CMC and STCMC surfaces in Riemannian and Lorentzian geometry.

De Sitter spacetime can be separated into two parts along two kinds of hypersurfaces and the half-de Sitter spacetimes are covered by the planar and hyperbolic coordinates respectively. Two positive energy theorems were proved previously for certain $¶$-asymptotically de Sitter and $\H$-asymptotically de Sitter initial…

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

We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of "lower energy" solutions to the above equation fails…

The paper analyzes flows related to Higgs energies on manifolds.

Study eternal solutions to Allen-Cahn equation on 3-sphere, connecting Clifford tori to equatorial spheres.