arXiv research

We prove a chord arc bound for disks embedded in $\mathbb{R}^3$ with constant mean curvature. This bound does not depend on the value of the mean curvature. It is inspired by and generalizes the work of Colding and Minicozzi in [2] for embedded minimal disks. Like in the minimal case, this chord arc bound is a fundamen…

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

Sharp chord-arc estimates for curve shortening flow on spheres.

In this paper we prove a theorem concerning lamination limits of sequences of compact disks $M_n$ embedded in $\mathbb{R}^3$ with constant mean curvature $H_n$, when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi…

In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [24] to prove to prove a weak chord arc type result for these disks…

The paper proves the existence and properties of geodesics on convex surfaces.

In this paper, we study the space of translational limits T(M) of a surface M properly embedded in R^3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface M' in T(M), the set T(M') in T(M). Among various dynamics type results we prove that surf…

The paper disproves the properness conjecture for higher-dimensional minimal hypersurfaces.

Study on minimal surfaces with constraints on index and branching order.

Study on helix curves and their Möbius energy asymptotics.