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Under a Morse index condition we prove symmetry results for solutions of a nonlinear mixed boundary condition elliptic problem. As an intermediate step we relate the Morse index of a solution to a mixed boundary condition linear eigenvalue…
Energy-minimizing constraint maps are a natural extension of the obstacle problem within a vectorial framework. Due to inherent topological constraints, these maps manifest a diverse structure that includes singularities similar to harmonic…
We extend to higher dimensions earlier sharp bounds for the area of two dimensional free boundary minimal surfaces contained in a geodesic ball of the round sphere. This follows work of Brendle and Fraser-Schoen in the euclidean case.
We construct random Morse functions on surfaces by random walk and compute related distributions. We study the space of Morse functions through these random variables. We consider subspaces characterized by the surfaces with boundary…
A very interesting problem in the classical theory of minimal surfaces consists of the classification of such surfaces under some geometrical and topological constraints. In this short paper, we give a brief summary of the known…
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…
Border's theorem gives an intuitive linear characterization of the feasible interim allocation rules of a Bayesian single-item environment, and it has several applications in economic and algorithmic mechanism design. All known…
In this paper, we establish strong holomorphic Morse inequalities on non-compact manifolds under the condition of optimal fundamental estimates. We show that optimal fundamental estimates are satisfied and then strong holomorphic Morse…
In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 1$ and $m\geq 1$ ideal boundary components. We show that the width for the area functional can be achieved by a bubble tree limit…
The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They…
In the paper, we study the Gauss map of a completely immersed anisotropic minimal surface with respect to convex parametric integrand in $\mathbb{R}^3$. By a local analysis, we prove the discreteness of the critical set of the Gauss map of…
In this paper, we give an improved Morse index bound of minimal hypersurfaces from Almgren-Pitts min-max construction in any closed Riemannian manifold $M^{n+1}$ $(n+1 \geq 3$), which generalizes a result by X. Zhou…
The concepts of exterior and boundary in multiset topological space are introduced. We further established few relationships between the concepts of boundary, closure, exterior and interior of an M- set. These concepts have been pigeonholed…
A convex geometry is finite zero-closed closure system that satisfies the anti-exchange property. Complexity results are given for two open problems related to representations of convex geometries using implication bases. In particular, the…
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 establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary…
This is an exposition of homotopical results on the geometric realization of semi-simplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The…
In earlier work we introduced topologically minimal surfaces as the analogue of geometrically minimal surfaces. Here we strengthen the analogy by showing that complicated amalgamations act as barriers to low genus, topologically minimal…
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…
In this paper we collect the main properties of free curves in the complex projective plane and a lot of conjectures and open problems, both old and new. In the quest to understand the mystery of free curves, many tools were developed and…