arXiv research

In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on Kähler manifolds. We first define a notion of weak solution of CSCK for an $L^\infty$ Kähler metric. The main result is to show that such a weak solution (with uniform $L^\infty$ bound…

Compactness results for hypersurfaces with mean curvature prescribed by an ambient function.

Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2 symmetry, establish the global well-posedness of the initial value problem for Einstein's field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity ass…

The paper finds the Finsler structure of Apollonian weak metric on unit disc.

We consider stationary autoregressive processes with coefficients restricted to an ellipsoid, which includes autoregressive processes with absolutely summable coefficients. We provide consistency results under different norms for the estimation of such processes using constrained and penalized estimators. As an applica…

We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.

Proves curvature tensor convergence for smoothable spaces.

Study proves existence of weak mean curvature flow with contact angle.

Proves inextendibility of weak null singularities from curvature blow-up.

We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notation for polygonal curves, where the polarity property is exploited, and then m…

We show that on compact Riemann surfaces of nonpositive curvature, the generalized periods, i.e. the $ν$-th order Fourier coefficients of eigenfunctions $e_λ$ over a closed smooth curve $γ$ which satisfies a natural curvature condition, go to 0 at the rate of $O((\logλ)^{-1/2})$, if $0<|ν|/λ<1-δ$, for any fixed $0<δ<1$…

New boundary condition for weak inverse mean curvature flow in bounded domains.

Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.

Surveying Ricci flow for weak lower scalar curvature bounds.

Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.

Researchers compute de Rham cohomology of geodesic flow foliations on hyperbolic surfaces.

Preserves scalar curvature bounds under weak convergence of 3-manifolds.

The study examines conditions for weak nearly cosymplectic manifolds to split into products.

We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.

Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.

Proves existence of proper solutions for inverse mean curvature flow.

The paper establishes bounds on scalar curvature on asymptotically flat manifolds.

Proves higher regularity for anisotropic inverse mean curvature flow.

Computed distortion coefficients for the α-Grushin plane.

Alternative proof of weak solutions to mean curvature flow using minimizing movements.

Novel weak solutions for volume-preserving mean curvature flow established.

We obtain the $C^{\a}$ regularity for weak solutions of a class of non-homogeneous ultraparabolic equation, with measurable coefficients. The result generalizes our recent $C^{\a}$ regularity results of homogeneous ultraparabolic equation.

We give a proof that Brakke's mean curvature flow under the unit density assumption is smooth almost everywhere in space-time. More generally, if the velocity is equal in a weak sense to its mean curvature plus some given α-Hölder continuous vector field, then we show C^{2,α} regularity almost everywhere.

Sharp regularity for Pfaff system leads to isometric immersions in arbitrary dimensions.

We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface $\partialΩ_0$, we show that there exists a weak solution to the null mean curvatu…

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…

New varifold solutions for mean curvature flow converge and are unique.

Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.

Paper proves mass theorems for nonnegative scalar curvature metrics.

We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula and the density up…

In this paper, we investigate special curves on a weak r-helix submanifold in Euclidean n-space E^{n}. Also, we give the important relations between weak r-helix submanifolds and the special curves such as line of curvature, asymptotic curve and helix line.

Study curvature properties of w.a. S-manifolds with new conditions.

Gradient estimates for solutions to a p-Laplacian equation on Riemannian manifolds.

Paper investigates conditions for independence of weak gradients on metric spaces.

Defines weak normals for irregular curves in high-dimensional spaces.

Develops weak formulation for spacelike flows in pseudo-Euclidean space.

We analyze the Standard & Poor's 500 stock market index from the last 22 years. The probability density function of price returns exhibits two well-distinguished regimes with self-similar structure: the first one displays strong super-diffusion together with short-time correlations, and the second one corresponds to we…

Study introduces weak elastic energy for curves on Riemannian surfaces.

The paper studies a new structure on contact manifolds and its foliation properties.

By making use of the nice behavior of Hawking masses of slices of a weak solution of inverse mean curvature flow in three dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean…

Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow preserves the area enclosed by the evolving curve, and this fact is used to prove a gene…

We propose a weak formulation for the binormal curvature flow of curves in $\R^3.$ This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence t…