Related papers: An enumerative min-max theorem for minimal surface…

In this paper, we construct and classify minimal surfaces foliated by horizontal constant curvature curves in product manifolds $M \times \R$, where $M$ is the hyperbolic plane, the Euclidean plane or the two dimensional sphere. The main…

We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson…

This paper is the first in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed Riemannian 3-manifold. The key for understanding such…

For every genus $g$, we prove that $S^2 \times R$ contains complete, properly embedded, genus-$g$ minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the $S^2$ tends to…

Let $\Sigma^2 \subset M^3$ be a minimal surface of index 0 or 1. Assume that a neighborhood of $\Sigma$ can be foliated by constant mean curvature (cmc) hypersurfaces. We use min-max theory and the catenoid estimate to construct…

We prove a parametrized compactness theorem on manifolds of bounded Ricci curvature, upper bounded diameter and lower bounded injectivity radius.

We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth $(n+1)$--dimensional Riemannian manifolds, a theorem proved first by Pitts for $2\leq n\leq 5$ and extended later by Schoen and Simon to…

In this paper, we use the conjugate surface construction to prove the existence of certain non-periodic symmetric immersed minimal surfaces. These surfaces have finite total curvature and embedded catenoid ends, and they have positive genus…

We classify minimal complex surfaces of general type with $p_g=q=3$. More precisely, we show that such a surface is either the symmetric product of a curve of genus 3 or a free $\Z_2-$quotient of the product of a curve of genus 2 and a…

We study min-max theory for area functional among hypersurfaces constrained in a smooth manifold with boundary. A Schoen-Simon-type regularity result is proved for integral varifolds which satisfy a variational inequality and restrict to a…

We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal…

This paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key is to understand the structure of an embedded minimal disk in a ball in…

An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total…

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum…

In this survey, we discuss various aspects of the minimal surface equation in the three-sphere S^3. After recalling the basic definitions, we describe a family of immersed minimal tori with rotational symmetry. We then review the known…

We give an algorithm to calculate the minimal and maximal genus of the orientable closed surface where a graph $G$ can be embedded. For this, we construct some special branched coverings of the 2-sphere. We apply this algorithm to calculate…

We prove a general fusion theorem for complete orientable minimal surfaces in $\mathbb{R}^3$ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are…

In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth,…

In this paper, we prove that every strictly convex 3-ball with nonnegative Ricci-curvature contains at least 3 embedded free-boundary minimal 2-disks for any generic metric, and at least 2 solutions even without genericity assumption. Our…

We prove that on a compact Riemannian manifold of dimension $3$ or higher, with positive Ricci curvature, the Allen--Cahn min-max scheme (implemented by the first author and N. Wickramasekera in 2020), with prescribing function taken to be…