arXiv research

We present here a result of Monomialization of real analytic two-symmetric tensor fields over regular real analytic surfaces. We apply it to the (extension of the pull-back of the) inner metric of a resolved surface of a real analytic surface singularity. Doing so we recover Hsiang & Pati property at each point of the …

Characterizes Bonnet surfaces using analytic conditions.

We analyze the slope gap distribution of Veech surfaces, finding finite non-analytic points and quadratic tail decay.

We show that for complex analytic K3 surfaces any torsion class in H^2(X,O_X^*) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surfa…

New method to parametrize infinite Riemann surfaces with bounded triangulations.

Interpolates curves using maximal and minimal surfaces in different spaces.

We prove that a maximal surface in Lorentz-Minkowski space $\Bbb L^3$ can be extended analytically along its boundary if the boundary lies in a plane meeting the surface at a constant angle.

New geometric proof shows index of umbilic points on analytic surfaces is at most one.

Analytic completeness criterion applied to constant mean curvature surfaces.

Using different forms of the arithmetic Riemann-Roch theorem and the computations of Bott-Chern secondary classes, we compute the analytic torsion and the height of Hirzebruch surfaces.

We classify torsion-free real-analytic affine connections on compact oriented real-analytic surfaces which are locally homogeneous on a nontrivial open set, without being locally homogeneous on all of the surface. In particular, we prove that such connections exist. This classification relies in a local result that cla…

This paper is about interpolating minimal surfaces between two real analytic curves, a and b, each of which are simple real analytic curves, using the Björling-Schwarz formula in the domain where it is valid, changing the normal distributions on inital curves. We insert curves $l_1,...l_L$ at specific locations and cla…

Analytic patch trees reveal new geometric structures and dimension fields.

We show that every real analytic action of a connected supersoluble Lie group on a compact surface with nonzero Euler characteristic has a fixed point. This implies that E. Lima's fixed point free $C^{\infty}$ action on $S^2$ of the affine group of the line cannot be approximated by analytic actions. An example is give…

Analytic curves linked to algebraic ones via Schottky groups.

Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.

Sphere eversions have been described so far by either pictures with minimal topological complexity, numerical evolution or complex equations. We write down relatively simple explicit formulas for the whole eversion, both analytic and topologically simpler, including also Boy surface (real projective plane), using a fam…

We prove that a real-valued function (that is not assumed to be continuous) on a real analytic manifold is analytic whenever all its restrictions to analytic submanifolds homeomorphic to the 2-sphere are analytic. This is a real analog for the classical theorem of Hartogs that a function on a complex manifold is comple…

This note provides a new proof of the real analyticity of the Liouville map.

Study projective connections on surfaces using osculating spaces.

We solve the Bonnet problem for surfaces in the homogeneous 3-manifolds with a 4-dimensional isometry group. More specifically, we show that a simply connected real analytic surface in H^2xR or S^2xR is uniquely determined pointwise by its metric and its principal curvatures if and only if it is not a minimal or a prop…

Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let $P = P_{g,k} = r_{g,k} \times R_+^k$ be the space of parabolic structures over Riemann surfac…

The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another complex analytic invariant called the harmonic volume. It is a natural extension o…

This research solves Plateau's problem for CRPC surfaces.

The Jorge-Meeks $n$-noid ($n\ge 2$) is a complete minimal surface of genus zero with $n$ catenoidal ends in the Euclidean 3-space $\boldsymbol{R}^3$, which has $(2π/n)$-rotation symmetry with respect to its axis. In this paper, we show that the corresponding maximal surface $f_n$ in Lorentz-Minkowski 3-space $\boldsymb…

We study analytic descriptions of conformal immersions of the Riemann sphere S^2 into the CP^(N-1) sigma model. In particular, an explicit expression for two-dimensional (2-D) surfaces, obtained from the generalized Weierstrass formula, is given. It is also demonstrated that these surfaces coincide with the ones obtain…

Proves no-hair theorem for certain vacuum black holes.

We prove the automorphic property of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion, in the case where the dimension of the moduli space is less than or equal to 2.

We develop analytical methods for nonlinear Dirac equations. Examples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds. We provide the key analytical steps, i.e., small energy regularity and removable…

Discrete approximation solves Björling's minimal surface problem.

We consider various metric and analytic notions of finiteness on translation surfaces. The Veech group of a surface is discrete if the surface has finite area or is totally bounded.

We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let $\nabla$ be …

In this article, we study an analog of the Björling problem for isothermic surfaces (that are more general than minimal surfaces): given a real analytic curve $γ$ in ${\mathbb R}^3$, and two analytic non-vanishing orthogonal vector fields $v$ and $w$ along $γ$, find an isothermic surface that is tangent to $γ$ and that…

In this paper we investigate the moduli space of parabolic Higgs bundles over a punctured Riemann surface with varying weights at the punctures. We show that the harmonic metric depends analytically on the weights and the stable Higgs bundle. This gives a Higgs bundle generalisation of a theorem of McOwen on the existe…

We extend the spectral theory of generalized Laplacians to integrable metrics on compact Riemann surfaces. As a consequence, we attach in a direct way, a holomorphic analytic torsion to any integrable metrics. We also provide a different approach to define the holomorphic analytic torsion. We prove that both approaches…

Study modular surfaces in Lorentz-Minkowski 3-space, classifying and analyzing their curvature and applications.

Paper connects hypersurfaces in 4D to surfaces in 3-sphere.

Let $a: I\to \mathbb{R}^3$ be a real analytic curve satisfying some conditions. In this article, we show that for any real analytic curve $l:I\to \mathbb R^3$ close to $a$ (in a sense which is precisely defined in the paper) there exists a translation of $l$, and a minimal surface which contains both $a$ and the tra…

We examine the space of surfaces in $\RR^{3}$ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the space $\Mk$ of all such surfaces with $k$ ends (where surfaces are identified if they differ by …

We calculate the free energy of Coulomb gas systems on Riemann surfaces.

In the present paper we study the Lie sphere geometry of Legendre surfaces by the method of moving frame and we prove an existence theorem for real-analytic Lie-minimal Legendre surfaces.

We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification, existence of normal forms, pencil/fibration decomposition, infinitesimal symmetries. We…

The period for a compact Riemann surface, defined by the integral of differential 1-forms, is a classical complex analytic invariant, strongly related to the complex structure of the surface. In this paper, we treat another complex analytic invariant called the pointed harmonic volume. As a natural extension of the per…

Symmetries of Einstein-Weyl manifolds can be extended from boundary surfaces.

Paper introduces a new metric for deforming surfaces with parabolics.

Let C be a real-analytic Jordan curve in $R^3$. Then C cannot bound infinitely many disk-type minimal surfaces which provide relative minima of area.

We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surface…

The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immer…