arXiv research

We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of $2 N$ dimensional Ricci flat (pseudo-) Riemannian geometries.

The study proves a theorem for surfaces using Codazzi operators and investigates parallel mean curvature surfaces.

We prove new local inequality for divisors on surfaces and utilize it to compute $α$-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$, $\mathbb{A}_{2}$, $\mathbb{A}_{3}$, $\mathbb{A}_{4}$, $\mathbb{A}_{5}$ or $\mathbb{A}_{6…

Constructs graphs with singularities in a special space.

Ray marching method visualizes flat surfaces efficiently.

Utilizing a splitting of geometric flows on surfaces introduced by Buzano and Rupflin, we present a general scheme to prove blow up criteria for such geometric flows. A vital ingredient is a new compactness theorem for families of metrics on surfaces with a uniform bound on their volumes, square integrals of their curv…

Study topological components of surface group representations into SL(2,R) and PSL(2,R).

Transforms curves and surfaces for efficient geometric analysis.

In the present paper we provide new examples of marginally trapped surfaces and tubes in FLRW spacetimes by using a basic relation between these objects and CMC surfaces in 3-manifolds. We also provide a new method to construct marginally trapped surfaces in closed FLRW spacetimes, which is based on the classical Hopf …

We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a uniqueness theorem and demonstrate that many knots cannot possess totally geodesic …

Paper calculates Morse index of Y-singular minimal surfaces.

We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3-manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to quantum Teichmuller space, or filtering the algebra and obtaining an associated graded algebra that…

Model uses Preisach hysteresis to predict gig worker acceptance, reducing costs and improving fill rates.

With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral surfaces with non-positive curvature. We show that in the case of such surfaces discr…

The paper finds new constant $p$-mean curvature surfaces in the Heisenberg group.

Study trisections on rational elliptic surfaces to find new Zariski pairs.

Decision-alignment evaluates uncertainty quantification for decision-relevant UQ

Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…

It is known that planar disks and small spherical caps are the only constant mean curvature graphs whose boundary is a round circle. Usually, the proof invokes the Maximum Principle for elliptic equations. This paper presents a new proof of this result motivated by an article due to Reilly. Our proof utilizes a flux fo…

Characterizes conformal classes of tori using differential geometry.

Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.

A new mosaic system for immersed surface-links is introduced.

Study transverse knots and symplectic surfaces using Seiberg-Witten monopole equations.

New method characterizes surface quadrilateral layouts as special immersions.

The paper studies constant harmonic mean curvature surfaces in Schwarzschild spaces, proving they foliate the space.

This paper classifies symmetries of biharmonic heat equations on surfaces of revolution.

We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the homotopy type of the infinite symmetric product of M^g, generalizing a well-known …

New analysis of crushing surfaces of positive genus impacts triangulation complexity.

There is an established bijection between finite-index subgroups Gamma of Gamma(2) and bipartite graphs on surfaces, or, equivalently, certain triples of permutations. We utilize this relationship to study both congruence and noncongruence subgroups in terms of the corresponding graphs. We show some elementary criteria…

New method for sensing non-planar surfaces using ERT.

Study of minimal surfaces in 4D with specific ends.

We propose and analyze sequential design methods for the problem of ranking several response surfaces. Namely, given $L \ge 2$ response surfaces over a continuous input space $\cal X$, the aim is to efficiently find the index of the minimal response across the entire $\cal X$. The response surfaces are not known and ha…

In this note, we provide a generalization for the definition of a trisection of a 4-manifold with boundary. We demonstrate the utility of this more general definition by finding a trisection diagram for the Cacime Surface, and also by finding a trisection-theoretic way to perform logarithmic surgery. In addition, we de…

The differential system for minimal Lagrangian surfaces in a $2_{\mathbb{C}}$-dimensional, non-flat, complex space form is an elliptic system defined on the bundle of oriented Lagrangian planes. This is a 6-symmetric space associated with the Lie group SL(3,$\mathbb{C}$), and the minimal Lagrangian surfaces arise as th…

Developed an ellipsoidal density-equalizing map for genus-0 closed surfaces.

This is an exposition of a proof of the Madsen-Weiss Theorem, which asserts that the homology of mapping class groups of surfaces, in a stable dimension range, is isomorphic to the homology of a certain infinite loopspace that arises naturally when one applies the "scanning method". The proof given here utilizes simpli…

The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.

Study uses sentiment analysis to predict implied volatility surface, improving prediction accuracy.

Authors calculate limits of curvatures on surfaces in sub-Riemannian manifolds.

In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4, then the distances on the surface are highly concentrated around the minimal poss…

Study optimizes CANN for actuarial tasks using RSM.

Uniform bound on geodesic images for surfaces using bicorn curves.

Networks, which represent agents and interactions between them, arise in myriad applications throughout the sciences, engineering, and even the humanities. To understand large-scale structure in a network, a common task is to cluster a network's nodes into sets called "communities", such that there are dense connection…

New subgroups of mapping class groups constructed for infinite-type surfaces.

Self-similar solutions to geometric flows are stable under small perturbations.

Study of surface defects in gauge theories leads to duality and separation of variables.

The paper studies Willmore surfaces in 4D conformal manifolds and finds the Clifford torus is strictly Willmore-stable.

A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in ${\mathbb E}^3$ must be greater than one. In this paper we prove this for $C^{3+α}$-smooth surfaces. The Conjecture is first reformulated in terms of complex points on a Lagrangian surface in $TS^2$, viewed as…