Related papers: Properness of associated minimal surfaces

Using min-max theory, we show that in any closed Riemannian manifold of dimension at least 3 and at most 7, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds…

We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $ N \times \mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N}…

The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is…

We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set $\mathcal{O} \subset F_n$ which is contained in the union of…

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…

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…

We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth $3$-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at…

We prove that every orientable infinite type surface without boundary and finite genus has a Riemann surface structure such that its modular group of quasiconformal homeomorphisms is countable.

We consider the isometric deformation problem for oriented non simply connected immersed minimal surfaces $f:M \to S^{4}$. We prove that the space of all isometric minimal immersions of $M$ into $S^{4}$ with the same normal curvature…

In previous work, the author proved that there is a countably infinite family of N=2 superconformal equivalence classes of DeWitt N=2 superconformal super-Riemann surfaces with closed, genus-zero body. In this paper, we determine the…

The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in $\R^{n+2}$ and those which are not conformally…

We show how to find a complete set of necessary and sufficient conditions that solve the fixed-parameter local congruence problem of immersions in $G$-spaces, whether homogeneous or not, provided that a certain $k^{\rm th}$ order jet bundle…

Near the end of his life, Bernhard Riemann made the marvelous discovery of a 1-parameter family $R_{\lambda}$, $\lambda\in (0,\infty)$, of periodic properly embedded minimal surfaces in $\mathbb{R}^3$ with the property that every horizontal…

If a piece of the contour of a picture is missing to the eye vision, then the brain tends to complete it using some kind of sub-Riemannian geodesics of the unit tangent bundle of the plane, R2xS1. These geodesics can be obtained by lifting…

Assume that $X$ is a connected, open, oriented smooth surface, $B$ is a compact Euclidean neighbourhood retract, and $\mathscr{J}=\{J_b\}_{b\in B}$ is a continuous family of complex structures on $X$ of local H\"older class…

We show that if $E$ is a closed convex set in $\mathbb C^n$ $(n>1)$ contained in a closed halfspace $H$ such that $E\cap bH$ is nonempty and bounded, then the concave domain $\Omega = \mathbb C^n\setminus E$ contains images of proper…

We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $\Sigma$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The…

In this paper we show for every sufficiently large integer $g$ the existence of a complete family of closed and embedded constant mean curvature (CMC) surfaces deforming the Lawson surfaces $\xi_{1,g}$ parametrized by their conformal type.…

A Ricci surface is a Riemannian 2-manifold $(M,g)$ whose Gaussian curvature $K$ satisfies $K\Delta K+g(dK,dK)+4K^3=0$. Every minimal surface isometrically embedded in $\mathbb{R}^3$ is a Ricci surface of non-positive curvature. At the end…

Minimal surfaces in a Riemannian manifold $M^n$ are surfaces which are stationary for area: the first variation of area vanishes. In this paper we focus on surfaces of the topological type of the real projective plane $\R P^2$. We show that…