Related papers: Tangential limits of stable minimal capillary surf…

In this paper we investigate the connection between the index and the geometry and topology of capillary surfaces. We prove an index estimate for compact capillary surfaces immersed in general 3-manifolds with boundary. We also study…

This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4\pi must be smoothly embedded. Related results are proved for varifolds and for soap film surfaces.

We develop a regularity and compactness theory for stable capillary minimal hypersurfaces in the half-space $\mathbb{H}^{n+1}$ with contact angle $\theta \in (0,\pi)$ and dimension $n \geq 2$. As a consequence, we obtain the generalized…

We establish a curvature estimate for classical minimal surfaces with total boundary curvature less than 4\pi. The main application is a bound on the genus of these surfaces depending solely on the geometry of the boundary curve. We also…

In this paper we prove that a capillary minimal surface outside the unit ball in $\mathbb {R}^3$ with one embedded end and finite total curvature must be either part of the plane or part of the catenoid. We also prove that a capillary…

In this paper, we study stability problem of anisotropic capillary hypersurfaces in an Euclidean half-space. We prove that any compact immersed anisotropic capillary constant anisotropic mean curvature hypersurface in the half-space is…

We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces…

We prove a local rigidity result for infinitesimally rigid capillary surfaces in some Riemannian $3$-manifolds with mean convex boundary. We also derive bounds on the genus, number of boundary components and area of any compact two-sided…

We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is…

In this paper we use stable capillary surfaces (analogous to the $\mu$-bubble construction) to study manifolds with strictly mean convex boundary and nonnegative scalar curvature. We give an obstruction to filling 2-manifolds by such…

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a…

We give improved estimates for the size of the singular set of minimizing capillary hypersurfaces: the singular set is always of codimension at least $4$, and this estimate improves if the capillary angle is close to $0$, $\frac{\pi}{2}$,…

We construct a complete, embedded minimal surface in euclidean 3-space which has unbounded Gaussian curvature. It has infinite genus, infinitely many catenoidal type ends and one limit end.

We investigate complete non-orientable minimal surfaces of finite total curvature in $\mathbb{R}^3$ such that their ends are foliated by closed lines of curvature. This condition on the ends is necessary if they have a piece inside some…

We develop minimal slicing via capillary hypersurfaces to understand positive scalar curvature metric on manifolds with boundary. The method provides rigidity statements once the regularity of minimizers of capillary area functional holds.…

In this paper we provide a large new family of embedded capillary surfaces inside polyhedral regions in the Euclidean space. The angle of contact of the examples we furnish is prescribed to be any value in $(\frac{\pi}{2}, \pi]$ and it is…

This is the first of two articles in which we investigate the geometry of free boundary and capillary minimal surfaces in balls $B_R\subset\mathbb{S}^3$. In this article, we extend our previous half-space intersection properties to warped…

We study stable immersed capillary hypersurfaces in a domain $\mathcal B$ which is either a half-space or a slab in the Euclidean space $\Bbb R^{n+1}.$ We prove that such a hypersurface $\Sigma$ is rotationally symmetric in the following…

In this paper, we introduce a new constrained mean curvature type flow for capillary boundary hypersurfaces in space forms. We show the flow exists for all time and converges globally to a spherical cap. Moreover, the flow preserves the…

In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the cylinder, we obtain some results of…