arXiv research

Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.

The paper generalizes free boundary min-max theory to equivariant settings.

The study proves the existence of free boundary minimal disks in convex regions.

The paper develops a theory for free boundary minimal surfaces with genus at least one.

The study finds at least 2 free-boundary minimal disks in convex 3-balls.

In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces which satisfy a uniform area bound. Our result is a natural generalization of the celebrated Schoen-Simon-Yau interior curvature estimates up to the free boundary. A direct corollary of our curvature estimates is…

In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory …

For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold with boundary $(M,\partial M)$ where $3\leq (n+1)\leq 7$, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates t…

Paper studies minimal surfaces in curved spaces, proving existence and properties.

In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, w…

Study boundary behavior of limit interfaces in Riemannian manifolds without convexity assumptions.

Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to ac…

Study anisotropic obstacle problem for minimal surfaces using Cahn-Hoffman transform.

The study finds minimal hypersurfaces in wedge-shaped manifolds with boundary.

Survey of rigidity and gap phenomena in sphere-ball submanifolds.

Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.

The paper studies the properties of maps with free boundaries, focusing on the obstacle case.

We obtain a series of results in the global theory of free boundary minimal surfaces, which in particular provide a rather complete picture for the way different complexity criteria, such as area, topology and Morse index compare, beyond the regime where effective estimates are at disposal.

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our met…

We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume…

Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.

In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish…

Constructs a unique surface in a ball with specific properties.

For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace-Beltrami operator on Riemannian manifold…

In this paper, we develop a general existence theory for properly embedded minimal surfaces with free boundary in any compact Riemannian 3-manifold $M$ with boundary $\partial M$. The main feature of our result is that no convexity assumption is required on $\partial M$. Our proof uses a variant of the min-max construc…

This work is the first step towards a description of the Gromov boundary of the free factor graph of a free product, with applications to subgroup classification for outer automorphisms. We extend the theory of algebraic laminations dual to trees, as developed by Coulbois, Hilion, Lustig and Reynolds, to the context of…

Study of free boundary minimal surfaces with new existence and degeneration results.

The paper proves the existence of boundary minimal hypersurfaces in compact manifolds with boundary.

In practical work with American put options, it is important to be able to know when to exercise the option, and when not to do so. In computer simulation based on the standard theory of geometric Brownian motion for simulating stock price movements, this problem is fairly easy to handle for options with a short lifesp…

Foundations for free boundary Brakke flows established.

We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula v…

Research proves the semi-classical limit of Liouville conformal field theory, describing deterministic geometry from random fluctuations.

Paper generalizes Minkowski inequality for umbilical hypersurfaces with free boundary.

We study multiplicity of constant scalar curvature metrics in products of a compact closed manifold and a compact manifold with boundary using equivariant bifurcation theory.

Estimates the growth of Morse index for free boundary minimal hypersurfaces.

In anomaly-free quantum field theories the integrand in the bosonic functional integral--the exponential of the effective action after integrating out fermions--is often defined only up to a phase without an additional choice. We term this choice ``setting the quantum integrand''. In the low-energy approximation to M-t…

In this paper we establish a connection between free boundary minimal surfaces in a ball in $\mathbb{R}^3$ and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.

Sharp geometric inequalities for free boundary hypersurfaces in balls.

The study finds conditions for free boundary Hamiltonian stationary discs in complex 2-space.

Study on metrics of surfaces with curvature and boundary conditions.

We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a free boundary equation. We also show that the value function is $C^1$ across the optimal stopping boundary. Our …

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

Optimal inequality for free boundary hypersurfaces in convex domains.

Proves existence of mean curvature flow with surgery for free boundary surfaces.

Study finds rigid properties of boundary-free hypersurfaces in specific data sets.

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

Connectivity proven in large rank Gromov boundary of free factor complex.

Constructs minimal annuli with free boundary in hyperbolic 3-space.