arXiv research

Critical surfaces are defined by Bachman as topological index 2 surfaces, generalizing incompressible surfaces and strongly irreducible surfaces. In this paper we give a condition to obtain critical Heegaard surfaces by amalgamation. As a special case, we obtain critical Heegaard surfaces by boundary stabilization. It …

Study classifies ruled surfaces critical to Dirichlet energy.

This paper studies symplectic critical surfaces in Hermite surfaces.

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 analogue to index 2 minimal surfaces, critical surfaces become quite natural.

Critical surfaces can be regarded as topological index 2 minimal surfaces which was introduced by David Bachman. In this paper we give a sufficient condition and a necessary condition for self-amalgamated Heegaard surfaces to be critical.

A closed, orientable, splitting surface in an oriented $3$-manifold is a topologically minimal surface of index $n$ if its associated disk complex is $(n-2)$-connected but not $(n-1)$-connected. A critical surface is a topologically minimal surface of index $2$. In this paper, we use an equivalent combinatorial definit…

In this paper we derive the Euler-Lagrange equation of the functional $L_β=\int_Σ\frac{1}{\cos^βα}dμ, ~~β\neq -1$ in the class of symplectic surfaces. It is $\cos^3α{\bf{H}}=β(J(J\nabla\cosα)^\top)^\bot$, which is an elliptic equation when $β\geq 0$. We call such a surface a $β$-symplectic critical surface. We first st…

This paper studies the critical dynamics of random surfaces, focusing on area and genus evolution.

Upper bounds found for systole function critical points on surface moduli space.

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 curvature surfaces in space forms. In this paper we investigate more generally the crit…

Simply connected surfaces with large constant mean curvature and free boundaries concentrate at critical points of the boundary's mean curvature.

Bound critical points for minimal Radó functions.

New foliations found for critical surfaces of Hawking energy, resolving discrepancies.

Surfaces in 3-manifolds concentrate at curvature critical points.

We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of intrinsic elliptic Lagrangians which are sub-critical or critical. They are derived using uniform $ε-$regularity estimates whic…

We show that the standard minimal genus Heegaard splitting of (closed orientable surface)\times S^1 is a critical Heegaard splitting.

In this paper we consider the compactness of $β$-symplectic critical surfaces in a Kähler surface. Let $M$ be a compact Kähler surface and $Σ_i\subset M$ be a sequence of closed $β_i$-symplectic critical surfaces with $β_i\toβ_0\in (0,\infty)$. Suppose the quantity $\int_{Σ_i}\frac{1}{\cos^qα_i}dμ_i$ (for some $q>4$) a…

Researchers compute trace formula for magnetic Laplacian on hyperbolic surfaces.

Free maps exist on low-dimensional tori and closed surfaces.

Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.

We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse's critical points. Using this method, we will investigate surface-knot with one critical point of index 1. Then we show infinitely many mutually distinct surface-knots that has an em…

The Hawking energy is nonnegative and rigid on area-constrained surfaces in general relativity.

In this paper are studied the simplest patterns of axial curvature lines (along which the normal curvature vector is at a vertex of the ellipse of curvature) near a critical point of a surface mapped into R4. These critical points, where the rank of the mapping drops from 2 to 1, occur isolated in generic one parameter…

In this paper we introduce "critical surfaces", which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an…

Investigates multifractal scaling in critical dynamics of random surfaces.

The paper classifies surfaces formed by quadrilateral gluings.

Proves stability of certain vector bundles on Kähler surfaces.

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…

We consider functions with isolated critical points on a closed surface. We prove that in a neighborhood of a critical point the function conjugates with Re$z^k$ for the some nonnegative integer k. The full topological invariant of such functions is constructed.

Paper proves Łojasiewicz inequalities near simple bubble trees on surfaces.

The paper examines critical points of solutions to a surface equation in 3D spacelike spaces.

Characterizes solutions to Z-critical equations on surfaces using effective conditions.

Study shows how electromagnetic and gravitational waves can form trapped surfaces.

The spinor representation is developed and used to investigate minimal surfaces in ${\bfR}^3$ with embedded planar ends. The moduli spaces of planar-ended minimal spheres and real projective planes are determined, and new families of minimal tori and Klein bottles are given. These surfaces compactify in $S^3$ to yield …

Paper derives second variation formula for eigenvalue functionals on surfaces.

The paper explores the shape of filling-systole subspace in surface moduli space and critical points of systole function.

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 critical point of the area functional with its Morse index as a critical point of t…

Classifies surfaces translating under specific curvature flows.

This paper studies properties of weak reducing pairs in critical Heegaard splittings.

Study of polynomial strata using braid groups and translation surfaces.

Constructs foliations of critical surfaces for Hawking energy in asymptotically flat initial data sets.

Paper proves trapped surface formation for EMCSF system without symmetry assumptions.

Study of critical tori for mean curvature energies in Killing submersions.

New mathematical surfaces without boundaries found.

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with nonzero mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch-Hawking mass. Th…

Improved bounds for eigenfunctions on hyperbolic surfaces found.

Numerically locating the critical points of non-convex surfaces is a long-standing problem central to many fields. Recently, the loss surfaces of deep neural networks have been explored to gain insight into outstanding questions in optimization, generalization, and network architecture design. However, the degree to wh…

We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent …