Related papers: CMC hypersurfaces with bounded Morse index

This paper is concerned with the completeness (with respect to the centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed in the ambient vector space. We show that completeness holds under generic regularity…

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in…

We develop a general structure theory for compact homogeneous Riemannian manifolds in relation to the co-index of symmetry. We will then use these results to classify irreducible, simply connected, compact homogeneous Riemannian manifolds…

We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with…

In 1977 P.Yang asked whether there exist complete immersed complex submanifolds g : M^k --> C^N with bounded image. A positive answer is known for holomorphic curves (k=1) and partial answers are known for the case when k>1. The principal…

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the…

In previous works, we have introduced the blown-up intersection cohomology and used it to extend Sullivan's minimal models theory to the framework of pseudomanifolds, and to give a positive answer to a conjecture of M. Goresky and W. Pardon…

In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish…

We consider a CMC hypersurface with an isolated singular point at which the tangent cone is regular, and such that, in a neighbourhood of said point, the hypersurface is the boundary of a Caccioppoli set that minimises the standard…

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 show that if C is a simple closed curve bounding an embedded disk in a closed 3-manifold M, then there exists a disk D in M with boundary C such that D minimizes the area among the embedded disks with boundary C. Moreover, D is smooth,…

In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of…

The techniques developed by Butscher in arXiv:math/0703469 for constructing constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere by gluing together spherical building blocks are generalized to handle less symmetric initial…

We present a simple inflationary scenario that can produce arbitrarily large spherical underdense or overdense regions embedded in a standard Lambda cold dark matter paradigm, which we refer to as bubbles. We analyze the effect such bubbles…

We show generic scarring phenomenon for minimal hypersurfaces in a class of complete non-compact manifolds. In particular, we prove that for any metric $g$ in a $C^{\infty}$-generic subset of the family of complete metrics which are thick…

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 show that the index of a constant mean curvature 1 surface in hyperbolic 3-space is completely determined by the compact Riemann surface and secondary Gauss map that represent it in Bryant's Weierstrass representation. We give three…

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 paper we are concerned with harmonic maps and minimal immersions defined on compact Riemannian manifolds and with values in homogenous strongly harmonic manifolds. We show some results on the Morse index by varying these maps along…

Let $\Sigma$ be a closed minimal surface immersed in a Riemannian 3-manifold carrying an orthonormal Killing frame. This class of ambient spaces includes Lie groups with a bi-invariant metric. In this paper, we prove that the sum of the…