arXiv research

The study counts Salem numbers linked to arithmetic hyperbolic orbifolds.

Paper provides a formula for translating solitons and singular minimal surfaces.

In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.

For every positive, continuous and homogeneous function $f$ on the space of currents on a compact surface $\overlineΣ$, and for every compactly supported filling current $α$, we compute as $L \to \infty$, the number of mapping classes $φ$ so that $f(φ(α))\leq L$. As an application, when the surface in question is close…

Based on a calibration argument, we prove a Bernstein type theorem for entire minimal graphs over Gauss space $\mathbb{G}^n$ by a simple proof.

Loewner inequality proven for curved surfaces.

We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally ramified value number for a wider class of complete minimal surfaces that includes algebraic minimal surfaces. It also provides a …

The paper studies how convex hypersurfaces in hyperbolic space evolve under a specific curvature flow.

In this short note we outline a simple probabilistic proof of the Gauss-Bonnet formula for compact Riemannian manifolds with boundary, which adapts to this setting an argument due to Hsu \cite{Hs1,Hs2} in the closed case. The new technical ingredient is the Feynman-Kac formula for differential forms satisfying absolute…

The $c$-curvature of a complete surface with Gauss curvature close to 1 in $C^2$ norm is almost-positive (in the sense of Kim--McCann). Our proof goes by a careful case by case analysis combined with perturbation arguments from the constant curvature case, keeping track of an estimate on the closeness curvature conditi…

This paper is devoted to $C^2$ a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature …

Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.

We establish existence of the eta-invariant as well as of the Atiyah-Patodi-Singer and the Cheeger-Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature, the Gauss-Bonnet and the spin Dirac operator. We derive an analogue of the Atiyah-Pato…

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the loca…

We compute the asymptotic growth rate of the number N(C, R) of closed geodesics of length less than R in a connected component C of a stratum of quadratic differentials. We prove that for any 0 < θ< 1, the number of closed geodesics of length at most R that spend at least θ-fraction of time outside of a compact subset …

New formula for knot invariants simplifies calculations and counts.

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^α K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. If $α\ge n+1$, we prove that the flow exists for …

Researchers solve a Plateau problem for maximal surfaces in pseudo-hyperbolic spaces.

We prove that the only closed, embedded ancient solutions to the curve shortening flow on $\mathbb{S}^2$ are equators or shrinking circles, starting at an equator at time $t=-\infty$ and collapsing to the north pole at time $t=0$. To obtain the result, we first prove a Harnack inequality for the curve shortening flow o…

The coamoeba of any complex algebraic plane curve $V$ is its image in the real torus under the argument map. The area counted with multiplicity of the coamoeba of any algebraic curve in $(\mathbb{C}^*)^2$ is bounded in terms of the degree of the curve. We show in this Note that up to multiplication by a constant in $(\…

We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isom…

Gauss diagrams' properties can change with Hamiltonian cycle choice.

By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically equivalent to the empty Gauss word, disproving a conjecture by Turaev. In fact, …

Algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.

We discuss Gauss codes of virtual diagrams and virtual doodles. The notion of a left canonical Gauss code is introduced and it is shown that oriented virtual doodles are uniquely presented by left canonical Gauss codes.

Incorrect parity-based descriptions of realizable Gauss diagrams found, but bipartite graphs provide a valid approach.

Study rotational surfaces with prescribed Gauss curvature in 3D space.

Paper solves Orlicz-Aleksandrov problem using Gauss curvature flow.

Study Gauss maps of surfaces in Heisenberg group using hyperbolic geometry.

Study on Gauss images of specific minimal surfaces with finite curvature.

Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.

We give an estimate of the Gauss curvature for minimal surfaces in ${\mathbb R}^m$ whose Gauss map omits more than $m(m+1)/2$ hyperplanes in ${\mathbb P}^{m-1}({\mathbb C})$.

The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.

The Gauss Image Measure uniquely identifies dual convex bodies up to dilation.

Study on multiple linking numbers, extending Gauss diagram formulas.

Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.

Study classifies hypersurfaces in 4D Lorentz-Minkowski space using specific operators.

We study the Gauss map of minimal surfaces in the Heisenberg group $\mathrm{Nil}_3$ endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane $\mathbb{H}^2$. Conversely, any nowhere antiholomorphic harmonic map into $\mathbb{…

New experiments show Gauss diagrams not all as simple as previously thought.

The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.

Classification of constant curvature surfaces in Berger spheres.

This foreword discusses the contributions of Bolyai, Gauss, and Lobachevsky to non-Euclidean geometry.

From the point of view of index theory, we give a simple proof of a Gauss-Bonnet-Chern formula for all Finsler manifolds by the Cartan connection. Based on this, we establish a Gauss-Bonnet-Chern formula for any metric-compatible connection and also derive the Gauss-Bonnet-Chern formula of Lackey.

We study the biharmonic stress-energy tensor $S_2$ of Gauss map. Adding few assumptions, the Gauss map with vanishing $S_2$ would be harmonic.

Gauss map of complete minimal surfaces avoids certain hypersurfaces.

We prove a global smooth isometric immersion for negatively curved surfaces with finite total curvature.

Generalizes Gauss-Bonnet to metrics with logarithmic singularities.

In this paper we use theory of embedded graphs on oriented and compact $PL$-surfaces to construct minimal realizations of signed Gauss paragraphs. We prove that the genus of the ambient surface of these minimal realizations can be seen as a function of the maximum number of Carter's circles. For the case of signed Gaus…