arXiv research

New approach to extremal hyperbolic surfaces using NEC groups.

Let $X$ be a compact Riemann surface of genus $\geq 2$ of constant negative curvature -1. An extremal disk is an embedded (resp. covering) disk of maximal (resp. minimal) radius. A surface containing an extremal disk is an {\em extremal surface}. This paper gives formulas enumerating extremal surfaces of genus $\geq 4$…

Extremal length systole is maximized at the Bolza surface.

Criterion found for Teichmüller extremal maps on infinite Riemann surfaces.

New maximal surfaces solve Bernstein problems.

Extremal spectral properties of Lawson tau-surfaces are investigated. The Lawson tau-surfaces form a two-parametric family of tori or Klein bottles minimally immersed in the standard unitary three-dimensional sphere. A Lawson tau-surface carries an extremal metric for some eigenvalue of the Laplace-Beltrami operator. U…

Non-convex extremal length found in surface metrics.

An extremal $k$-packing is a collection of $k$ mutually disjoint metric discs, embedded in a surface, whose radius is maximal for the given topology. We study compact non-orientable surfaces of genus $g\ge 3$ containing extremal $k$-packings.

In this paper, we are concerned with light-like extremal surfaces in curved spacetimes. It is interesting to find that under a diffeomorphic transformation of variables, the light-like extremal surfaces can be described by a system of nonlinear geodesic equations. Particularly, we investigate the light-like extremal su…

Study classifies special metrics on specific surfaces.

We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil…

We prove an optimal systolic inequality for nonpositively curved Dyck's surfaces. The extremal surface is flat with eight conical singularities, six of angle theta and two of angle 9pi - theta, for a suitable theta with cos(theta) in Q(sqrt{19}). Relying on some delicate capacity estimates, we also show that the extrem…

The aim of this article is to study expansions of solutions to an extremal metric type equation on the blow-up of constant scalar curvature Kähler surfaces. This is related to finding constant scalar curvature Kähler (cscK) metrics on K-stable blow-ups of extremal Kähler surfaces

Study on black holes and photon surfaces in 4D spacetimes, proving uniqueness theorems.

Paper shows mapping class groups are not extremely amenable except for specific cases.

Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann surfaces to $\mathbb{R}$-trees, we study the second variation of extremal length fu…

We consider smoothings of a complex surface with singularities of class T and no nontrivial holomorphic vector field. Under an hypothesis of non degeneracy of the smoothing at each singular point, we prove that if the singular surface admits an extremal metric, then the smoothings also admit extremal metrics in nearby …

Study confirms conjecture on extremal length of hyperbolic metrics.

Motivated by the HRRT-formula for holographic entanglement entropy, we consider the following question: what are the position and the surface area of extremal surfaces in a perturbed geometry, given their anchor on the asymptotic boundary? We derive explicit expressions for the change in position and surface area, ther…

New examples of extremal Kähler metrics on blow-ups of parabolic ruled surfaces are constructed. The method is based on the gluing construction of Arezzo, Pacard and Singer. This enables to endow ruled surfaces of the form $\mathbb{P}(\mathcal{O}\oplus L)$ with special parabolic structures such that the associated iter…

Extremal compact hyperbolic surfaces contain a packing of discs of the largest possible radius permitted by the topology of the surface. It is well known that arithmetic conditions on the uniformizing group are necessary for the existence of a second extremal packing in the same surface, but constructing explicit examp…

The paper introduces twins in Kähler and Sasaki geometry, generalizing known concepts.

In this paper, we study the Abreu equation on toric surfaces. In particular, we prove the existence of the positive extremal metric when relative $K$-stability is assumed.

The paper proves the existence of a special Kähler metric on a minimal ruled surface.

The paper develops a continiuty method for solutions of the Abreu equation, which include extremal metrics on toric surfaces. Results are obtained, assuming a hypothesis (the "M-condition") on the solutions.

New proof shows rationality of scl for non-filling curves.

We give some detailed numerical information about extremal metrics on four different toric surfaces. These are sample of many other cases which can be treated using a computer programme outlined in the paper.

The i-th eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of fixed area. Extremal points of these functionals correspond to surfaces admitting minimal isometric immersions into spheres. Recently, critical metrics for the first eigenvalue were classified on tori a…

Study finds Scherk type surfaces as extremals for zero-curvature minimal graphs.

Proves uniqueness of certain spacetime solutions with extremal horizons.

The $i$-th eigenvalue $λ_i$ of the Laplace-Beltrami operator on a surface can be considered as a functional on the space of all Riemannian metrics of unit volume on this surface. Surprisingly only few examples of extremal metrics for these functionals are known. In the present paper a new countable family of extremal m…

The regularity of systolically extremal surfaces is a notoriously difficult problem already discussed by M. Gromov in 1983, who proposed an argument toward the existence of $L^2$-extremizers exploiting the theory of $r$-regularity developed by P. A. White and others by the 1950s. We propose to study the problem of syst…

The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.

We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $Γ$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\…

Study wormholes in surface moduli space, proving conjecture.

We prove that a Lefschetz fibration over the disc that, after compactification, has the same singular fibers as an extremal rational elliptic surface can be obtained by deleting a singular fiber and a section from the rational extremal elliptic surface, i.e. such a Lefschetz fibration is determined up to topological eq…

Extending BTZ models to complete hyperbolic surfaces.

Recently it was shown that the area A and the angular momentum J of any apparent horizon on a maximal, axisymmetric and asymptotically flat Cauchy hyper-surface of a vacuum space-time satisfy necessarily the universal inequality A >= 8 pi |J|. We show here that the equality A=8 pi |J| is never attained. As equality is …

Let $T$ be a triangulation of a Riemann surface. We show that the 1-skeleton of $T$ may be oriented so that there is a global bound on the outdegree of the vertices. Our application is to construct extremal metrics on triangulations formed from $T$ by attaching new edges and vertices and subdividing its faces. Such ref…

Survey on Thurston metric on Teichmüller space, focusing on extremal maps.

We apply recently developed convex programs to find the minimal-area Riemannian metric on $2n$-sided polygons ($n\geq 3$) with length conditions on curves joining opposite sides. We argue that the Riemannian extremal metric coincides with the conformal extremal metric on the regular $2n$-gon. The hexagon was considered…

In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we conjecture to be equivalent to the existence of an extremal metric in the polaris…

We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of M. Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermo…

Sharp inequalities and extremals on compact Riemann surfaces with boundary.

Extends Newton's minimal resistance problem to Riemannian surfaces.

The paper shows how to unknot certain nonorientable surfaces in 4-dimensional spaces.

The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus.…

The study finds no extremal metrics for eigenvalues on compact manifolds but constructs examples for annuli.