arXiv research

Study shows convergence of volumes on manifolds with boundary under area constraints.

Study of Ricci flow convergence on surfaces with boundary.

The paper studies free boundary minimal surfaces with many boundaries and their convergence to closed minimal surfaces.

The paper proves conditions for smooth convergence of hyperkaehler 4-manifolds with boundary.

We prove a convergence theorem on the moduli space of constant $σ_{2}$ metrics for conic 4-spheres. We show that when a numerical condition is convergent to the boundary case, the geometry of conic 4-spheres converges to the boundary case while preserving capacity.

Paper proves convergence of MDL to Einstein-Hilbert with boundary term.

We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into th…

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

The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.

The paper proves stability of manifolds with boundary under volume and distance constraints.

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

The paper solves a flow problem on surfaces with boundary to converge to the hyperbolic metric.

We study the Yamabe flow on compact Riemannian manifolds of dimensions greater than two with minimal boundary. Convergence to a metric with constant scalar curvature and minimal boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin.

Holomorphic discs converge to maximal surfaces under specific flows.

New method shows Hessian estimator from random samples converges to true Hessian on complex manifolds.

PINN-FEM combines PINNs and FEM for accurate Dirichlet boundary condition enforcement.

We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that it converges to a hemisphere, when t…

Study on curve shortening flow with boundary conditions, proving convergence or contraction.

The paper develops a theory of conformal density at infinity for groups with contracting elements.

Proves convergence groups on a 2-sphere are Kleinian groups.

We study a conformal flow for compact Riemannian manifolds of dimension greater than two with boundary. Convergence to a scalar-flat metric with constant mean curvature on the boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin or if it satisfies a generic condition.

We establish a Schn$\ddot{\text{u}}$rer's convergence result and then apply it to obtain the existence of solutions on the second boundary value problem for a family of special Lagrangian equations

New algorithm improves convergence for non-convex problems with boundaries.

Study ancient solutions to free boundary mean curvature flow in convex manifolds.

Analyzes Willmore flow for graphs with boundary data, proving existence and convergence.

A new boundary for geodesic spaces defined and studied.

Study of group boundaries and subgroup properties.

Study shows how a curve shortens to a half-circle under specific flow.

Topology of non-orientable spaces without boundary is studied.

This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric c…

The paper proves Ricci flow convergence on compact manifolds with boundary.

Improved fourth-order compact scheme for option valuation with Robin boundary condition.

We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and deri…

Rare Teichmüller disks converge to small limit sets.

We study Hawking mass and the Huisken's isoperimetric mass evaluated on surfaces with boundary. The convergence to an ADM mass defined on asymptotically flat manifold with a non-compact boundary are proved.

We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian …

In a recent paper A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces $Σ\_n$ in $B^3$ which have genus $0$ and $n$ boundary components, for all $n \geq 3$. For large $n$, we give an independent construction of $Σ\_n$ and prove the existence of free boundary minimal surfaces $\tilde Σ\_n…

Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical convergence away from finitely many points. We show that the limit of a sequence …

Recall that Federer-Fleming defined the notion of flat convergence of submanifolds of Euclidean space to solve the Plateau problem. Here we prove the upper semicontinuity of Neumann eigenvalues of the submanifolds when they converge in the flat sense without losing volume. With an additional condition on the boundaries…

Let $X$ be a proper CAT(0) space and let $G$ be a cocompact group of isometries of $X$ which acts properly discontinuously. Charney and Sultan constructed a quasi-isometry invariant boundary for proper CAT(0) spaces which they called the contracting boundary. The contracting boundary imitates the Gromov boundary for $δ…

We consider a fully nonlinear parabolic equation with nonlinear Neumann type boundary condition, and show that the longtime existence and convergence of the flow. Finally we apply this study to the boundary value problem for minimal Lagrangian graphs.

Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.

We study the free boundary Euler equations in two spatial dimensions. We prove that if the boundary is sufficiently regular, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity.

Study on metric spaces with properties (ETR), (LBD) and their convergence.

This article is a continuation of earlier work [R.L. Huang and Y.H. Ye, On the second boundary value problem for a class of fully nonlinear flows I, to appear in International Mathematics Research Notices], where the long time existence and convergence were given on some general parabolic type special Lagrangian equati…

We prove Cheeger-Gromov convergence for a subsequence of a given sequence of manifolds-with-boundary of bounded geometry. The method of the proof is to reduce, via height functions, the problem to the setting of Hamilton's compactnes theorem for manifolds without boundary.

We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient di…

We introduce a deformation of Riemann surfaces and we are interested in the convergence of this deformation to a point of the Gardiner-masur boundary of Teichmueller space. This deformation, which we call the horocyclic deformation, is directed by a projective measured foliation and belongs to a certain horocycle in a …