Related papers: On the existence of high index topologically minim…

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…

We study the topological index of some irregular surfaces that we call generalized Lagrangian. We show that under certain hypotheses on the base locus of the Lagrangian system the topological index is non-negative. For the minimal surfaces…

The disk complex of a surface in a 3-manifold is used to define its {\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical…

In this paper, we show the existence of a timelike minimal surface with an arbitrary number of weak complete ends. Then, we discuss the asymptotic behaviour of the simple ends and the topology of the singularity set of the constructed…

We show that for an immersed two-sided minimal surface in $R^3$, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in $R^3$ of index $2$, as…

This paper contains the motivation for the study of critical surfaces. In previous work the only justification given for the definition of this new class of surfaces is the strength of the results. However, when viewed as the topological…

We show that in a closed 3-manifold with a generic metric of positive Ricci curvature, there are minimal surfaces of arbitrary large Morse index, which partially confirms a conjecture by F. Marques and A. Neves. We prove this by analyzing…

For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously…

We show that a 3-manifold containing an incompressible surface has topologically minimal surfaces of arbitrary high genus.

We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of…

We show that there are a finite number of possible pictures for a surface in a tetrahedron with local index $n$. Combined with previous results, this establishes that any topologically minimal surface can be transformed into one with a…

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…

Recently a simple proof of the generalizations of Hawking's black hole topology theorem and its application to topological black holes for higher dimensional ($n\geq 4$) spacetimes was given \cite{rnew}. By applying the associated new line…

Topological photonics provides a powerful framework to describe and understand many nontrivial wave phenomena in complex electromagnetic platforms. The topological index of a physical system is an abstract global property that depends on…

This paper gives, in generic situations, a complete classification of ruled minimal surfaces in pseudo-Euclidean space with arbitrary index. In addition, we discuss the condition for ruled minimal surfaces to exist, and give a…

This is an expository paper which presents the holomorphic classification of rational complex surfaces from a simple and intuitive point of view, which is not found in the literature. Our approach is to compare this classification with the…

We present a new topological proof of the infinitude of prime numbers with a new topology. Furthermore, in this topology, we characterize the infinitude of any non-empty subset of prime numbers.

We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via…

We prove a compactness result for minimal hypersurfaces with bounded index and volume, which can be thought of as an extension of the compactness theorem of Choi-Schoen (Invent. Math. 1985) to higher dimensions.

We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are…