Related papers: Index Estimates for CMC and Minimal Surfaces with …
We prove a linear upper bound on the Morse index of closed constant mean curvature (CMC) surfaces in orientable three-manifolds in terms of genus, number of branch points and a Willmore-type energy.
We prove index estimates for closed and free boundary CMC surfaces in certain $3$-dimensional submanifolds of some Euclidean space. When the mean curvature is large enough we are able to prove that the index of a CMC surface in an arbitrary…
This is the second paper in our sequence. Here, we apply our abstract Morse index formulation developed in the previous paper to study several optimization set-ups with constraints, including type I or/and type II considerations. A common…
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 is the second 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 find monotonicity formulae which imply that capillary minimal…
Using the local picture of the degeneration of sequences of minimal surfaces developed by Chodosh, Ketover and Maximo we show that in any closed Riemannian 3-manifold $(M,g)$, the genus of an embedded CMC surface can be bounded only in…
In this paper, we consider compact free boundary constant mean curvature surfaces immersed in a mean convex body of the Euclidean space or in the unit sphere. We prove that the Morse index is bounded from below by a linear function of the…
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…
The conformal parameterisation of a minimal surface is harmonic. Therefore, a minimal surface is a critical point of both the energy functional and the area functional. In this paper, we compare the Morse index of a minimal surface as a…
Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above…
We prove that the space of free boundary CMC surfaces of bounded topology, bounded area and bounded boundary length is compact in the $C^k$ graphical sense away from a finite set of points. This is a CMC version of a result for minimal…
This article investigates stationary surfaces with boundaries, which arise as the critical points of functionals dependent on curvature. Precisely, a generalized "bending energy" functional $\mathcal{W}$ is considered which involves a…
We develop a bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular we show that convergence always occurs with multiplicity one, which…
We consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index - the number of…
Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy $W=\int H^2$ under compactly supported infinitesimal conformal variations. Examples include all constant mean…
Defined mathematically as critical points of surface area subject to a volume constraint, constant mean curvatures (CMC) surfaces are idealizations of interfaces occurring between two immiscible fluids. Their behavior elucidates phenomena…
In this note we consider asymptotically flat manifolds with non-negative scalar curvature and an inner boundary which is an outermost minimal surface. We show that there exists an upper bound on the mean curvature of a constant mean…
In this sequence, we first prove an abstract Morse index theorem in a Hilbert space modeling a variational problem with constraints. Then, our abstract formulation is applied to study several optimization setups including closed CMC…
In this paper, we establish monotonicity formulas for capillary surfaces in the half-space $\mathbb{R}^3_+$ and in the unit ball $\mathbb{B}^3$ and extend the result of Volkmann (Comm. Anal. Geom.24(2016), no.1, 195~221.…
Given $r_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq 0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq r_0$ and with the supremum of absolute sectional curvature at most $K_0$, and let…