arXiv research

Study on closed curves on negatively curved surfaces, linking number formula, and restrictions.

Veering branched surfaces help construct geodesic flows on curved surfaces.

New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.

The existence of essential closed surfaces surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.

Study curve shortening flows on specific surfaces, proving properties and existence.

Study finds negatively curved spheres in elliptic surfaces and their modifications.

In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.

The paper constructs foliations of minimal surfaces in negatively curved 3-manifolds.

We introduced an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature less than or equal -1, by the hyperbolic metric.

We prove by an algebraic method that the embedding of the Teichmuller space in the space of geodesic currents is totally linearly independent. We prove a similar result for all negatively curved surfaces using an ergodic argument.

The Poincaré series for surfaces with boundary extends to the complex plane.

If a piece of the contour of a picture is missing to the eye vision, then the brain tends to complete it using some kind of sub-Riemannian geodesics of the unit tangent bundle of the plane, R2xS1. These geodesics can be obtained by lifting extremal curves of a total curvature type energy in the plane. We completely sol…

It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting …

We obtain an estimate from below for the remainder in Weyl's law on negatively curved surfaces. In the constant curvature case, such a bound was proved independently by Hejhal and Randol in 1976 using the Selberg zeta function techniques. Our approach works in arbitrary negative curvature, and is based on wave trace as…

Study on moduli spaces of negatively curved metrics on surfaces.

Length spectral rigidity is the question of under what circumstances the geometry of a surface can be determined, up to isotopy, by knowing only the lengths of its closed geodesics. It is known that this can be done for negatively curved Riemannian surfaces, as well as for negatively-curved cone surfaces. Steps are tak…

New Teichmüller space for negatively curved surfaces defined.

The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.

Study on $k$-surfaces in negatively curved 3-manifolds, focusing on energy and entropy.

Improved Strichartz estimates for Schrödinger equation on negatively curved manifolds.

We prove the existence of $C^{1,1}$ isometric immersions of several classes of metrics on surfaces $(\mathcal{M},g)$ into the three-dimensional Euclidean space $\mathbb{R}^3$, where the metrics $g$ have strictly negative curvature. These include the standard hyperbolic plane, generalised helicoid-type metrics and gener…

In this note, we prove a Schwarz-Pick type lemma for minimal maps between negatively curved Riemannian surfaces. More precisely, we prove that if $f:M \to N$ is a minimal map with bounded Jacobian between two complete negatively curved Riemann surfaces M and N whose sectional curvatures $σ_M$ and $σ_N$ satisfy $infσ_M …

Hilbert-Efimov theorem states that any complete surface with curvature bounded above by a negative constant can not be isometrically imbedded in $\mathbb{R}^3.$ We demonstrate that any simply-connected smooth complete surface with curvature bounded above by a negative constant admits a smooth isometric embedding into t…

New proof shows surfaces can have identical length spectra but not simple ones.

Characterizes unknotted curves on Seifert surfaces of twist knots.

The flat trace of geodesic Koopman operators varies with negatively curved surfaces.

The study shows how certain surfaces can be filled by hyperbolic manifolds.

Study shows negatively curved manifolds' spherical volume equals minimal surface area.

New counterexample shows curved surfaces can deform geodesics without diffeomorphism.

Researchers prove unique embedding of curved surfaces into Minkowski spacetime.

We show uniqueness of Ricci flows starting at a surface of uniformly negative curvature, with the assumption that the flows become complete instantaneously. Together with the more general existence result proved in [10], this settles the issue of well-posedness in this class.

Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus $n$ with $k$ punctures for any $n$ and $k$.

We constructed physically stable sp2 negatively curved cubic carbon structures which reticulate a Schwarz P-like surface. The method for constructing such crystal structures is based on the notion of the standard realization of abstract crystal lattices. In this paper, we expound on the mathematical method to construct…

In this note, we develop a condition on a closed curve on a surface or in a 3-manifold that implies that the curve has the property that its length function on the space of all hyperbolic structures on the surface or 3-manifold completely determines the curve. For an orientable surface $S$ of negative Euler characteris…

Estimates the number of closed curves on surfaces with power-saving error terms.

Paper proves existence of isometric immersions for negatively curved surfaces with unbounded second fundamental form.

We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold $N$ with sectional curvatures bounded from above by a negative quantity $K_{N}\leq b<0$

The entropy of minimal surfaces is minimized in hyperbolic manifolds.

One-harmonic maps from a curved surface to hyperbolic plane have specific interior properties.

Let $X$ be an irreducible smooth geometrically integral projective surface over a field. In this paper we give an effective bound in terms of the Neron--Severi rank $ρ(X)$ of $X$ for the number of irreducible curves $C$ on $X$ with negative self-intersection and geometric genus less than $b_1(X)/4$, where $b_1(X)$ is t…

We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length at most $L$ grows exponentially in $L$. We get exponentially tighter bounds given…

Study finds geodesic networks for surfaces with convex boundary.

Let $X, Y$ be complete, simply connected Riemannian surfaces with pinched negative curvature $-b^2 \leq K \leq -1$. We show that if $f : \partial X \to \partial Y$ is a Moebius homeomorphism between the boundaries at infinity of $X, Y$, then $f$ extends to an isometry $F : X \to Y$. This can be viewed as a generalizati…

Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved Kähler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex surfaces may share the rigidity of quaternionic/octionic hyperbolic manifolds, ou…

Let $(M,g)$ be a closed oriented negatively curved surface. A unitary connection on a Hermitian vector bundle over $M$ is said to be transparent if its parallel transport along the closed geodesics of $g$ is the identity. We study the space of such connections modulo gauge and we prove a classification result in terms …

We derive a lower bound to the spectral threshold of the Dirichlet Laplacian in tubular neighbourhoods of constant radius about complete surfaces. This lower bound is given by the lowest eigenvalue of a one-dimensional operator depending on the radius and principal curvatures of the reference surface. Moreover, we show…

New metrics connect surfaces with Anosov flows to those with negative curvature.