scholarly journals Spectral properties and conformal type of surfaces

2002 ◽  
Vol 74 (4) ◽  
pp. 585-588 ◽  
Author(s):  
PHILIPPE CASTILLON
Keyword(s):  
Riemann Surface ◽  
Finite Number ◽  
Minimal Surfaces ◽  
Spectral Properties ◽  
Short Note ◽  
Riemannian Surface ◽  
Curvature Function ◽  
Surface Laplacian ◽  
Scalar Multiple

In this short note, we announce a result relating the geometry of a riemannian surface to the positivity of some operators on this surface (the operators considered here are of the form surface Laplacian plus a scalar multiple of the curvature function). In particular we obtain a theorem "à la Huber'': under a spectral hypothesis we prove that the surface is conformally equivalent to a Riemann surface with a finite number of points removed. This problem has its origin in the study of stable minimal surfaces.

scholarly journals Embedded minimal surfaces of finite topology

2019 ◽  
Vol 2019 (753) ◽  
pp. 159-191 ◽  
Author(s):  
William H. Meeks III ◽  
Joaquín Pérez
Keyword(s):  
Asymptotic Behavior ◽  
Riemann Surface ◽  
Finite Number ◽  
Minimal Surface ◽  
Minimal Surfaces ◽  
Compact Riemann Surface ◽  
Finite Topology ◽  
Compact Boundary ◽  
Interior Points

AbstractIn this paper we prove that a complete, embedded minimal surface M in {\mathbb{R}^{3}} with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface {\overline{M}} with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion {\overline{M}}. In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of M.

Invariant Subspaces on Riemann Surfaces

1966 ◽  
Vol 18 ◽  
pp. 399-403 ◽  
Author(s):  
Michael Voichick
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Finite Number ◽  
Harmonic Measure ◽  
Riemann Surfaces ◽  
Invariant Subspaces ◽  
Riemann Surface With Boundary ◽  
Analytic Curves

In this paper we generalize to Riemann surfaces a theorem of Helson and Lowdenslager in (2) describing the closed subspaces of L2(﹛|z| = 1﹜) that are invariant under multiplication by eiθ.Let R be a region on a Riemann surface with boundary Γ consisting of a finite number of disjoint simple closed analytic curves such that R ⋃ Γ is compact and R lies on one side of Γ. Let dμ be the harmonic measure on Γ with respect to a fixed point t0 on R. We shall consider the closed subspaces of L2(Γ, dμ) that are invariant under multiplication by functions in A (R) = ﹛F|F continuous on , analytic on R}.

Covering properties of open continuous mappings having two valences between Riemann surfaces

1995 ◽  
Vol 118 (2) ◽  
pp. 321-340 ◽  
Author(s):  
Abdallah Lyzzaik
Keyword(s):  
Riemann Surface ◽  
Finite Number ◽  
Riemann Surfaces ◽  
Embedding Theorem ◽  
Branch Points ◽  
Continuous Mappings ◽  
Counting Multiplicity ◽  
Covering Properties ◽  
Closed Riemann Surface ◽  
Finite Genus

AbstractLet be an open Riemann surface with finite genus and finite number of boundary components, and let be a closed Riemann surface. An open continuous function from to is termed a (p, q)-map, 0 < q < p, if it has a finite number of branch points and assumes every point in either p or q times, counting multiplicity, with possibly a finite number of exceptions. These comprise the most general class of all non-trivial functions having two valences between and .The object of this paper is to study the geometry of (p, q)-maps and establish a generalized embedding theorem which asserts that the image surfaces of (p, q)-maps embed in p-fold closed coverings possibly having branch points off the image surfaces.

scholarly journals Monotone homotopies and contracting discs on Riemannian surfaces

2018 ◽  
Vol 10 (02) ◽  
pp. 323-354 ◽  
Author(s):  
Gregory R. Chambers ◽  
Regina Rotman
Keyword(s):  
Finite Volume ◽  
Minimal Surfaces ◽  
Analogous Result ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Riemannian Surface ◽  
Closed Curves ◽  
Minimal Hypersurfaces ◽  
Riemannian Surfaces ◽  
Complete Manifolds

A monotone homotopy is a homotopy composed of simple closed curves which are also pairwise disjoint. In this paper, we prove a “gluing” theorem for monotone homotopies; we show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem are used in [G. R. Chambers and Y. Liokumovich, Existence of minimal hypersurfaces in complete manifolds of finite volume, arXiv:1609.04058] to prove an analogous result for cycles, which forms a critical step in their proof of the existence of minimal surfaces in complete non-compact manifolds of finite volume. We also show that, if monotone homotopies exist, then fixed point contractions through short curves exist. In particular, suppose that [Formula: see text] is a simple closed curve of a Riemannian surface, and that there exists a monotone contraction which covers a disc which [Formula: see text] bounds consisting of curves of length [Formula: see text]. If [Formula: see text] and [Formula: see text], then there exists a homotopy that contracts [Formula: see text] to [Formula: see text] over loops that are based at [Formula: see text] and have length bounded by [Formula: see text], where [Formula: see text] is the diameter of the surface. If the surface is a disc, and if [Formula: see text] is the boundary of this disc, then this bound can be improved to [Formula: see text].

scholarly journals Bakry–Émery Curvature Functions on Graphs

2019 ◽  
Vol 72 (1) ◽  
pp. 89-143 ◽  
Author(s):  
David Cushing ◽  
Shiping Liu ◽  
Norbert Peyerimhoff
Keyword(s):  
Lower Bound ◽  
Spectral Properties ◽  
Cayley Graphs ◽  
Cartesian Product ◽  
Upper And Lower Bounds ◽  
Curvature Function ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Local Structures ◽  
Local Properties

AbstractWe study local properties of the Bakry–Émery curvature function ${\mathcal{K}}_{G,x}:(0,\infty ]\rightarrow \mathbb{R}$ at a vertex $x$ of a graph $G$ systematically. Here ${\mathcal{K}}_{G,x}({\mathcal{N}})$ is defined as the optimal curvature lower bound ${\mathcal{K}}$ in the Bakry–Émery curvature-dimension inequality $CD({\mathcal{K}},{\mathcal{N}})$ that $x$ satisfies. We provide upper and lower bounds for the curvature functions, introduce fundamental concepts like curvature sharpness and $S^{1}$-out regularity, and relate the curvature functions of $G$ with various spectral properties of (weighted) graphs constructed from local structures of $G$. We prove that the curvature functions of the Cartesian product of two graphs $G_{1},G_{2}$ are equal to an abstract product of curvature functions of $G_{1},G_{2}$. We explore the curvature functions of Cayley graphs and many particular (families of) examples. We present various conjectures and construct an infinite increasing family of 6-regular graphs which satisfy $CD(0,\infty )$ but are not Cayley graphs.

HARMONIC MORPHISMS AND HERMITIAN STRUCTURES ON EINSTEIN 4-MANIFOLDS

1992 ◽  
Vol 03 (03) ◽  
pp. 415-439 ◽  
Author(s):  
JOHN C. WOOD
Keyword(s):  
Riemann Surface ◽  
Critical Points ◽  
Minimal Surfaces ◽  
Hermitian Structure ◽  
Harmonic Morphisms ◽  
Harmonic Morphism ◽  
Degeneracy Condition ◽  
Hermitian Structures

We show that a submersive harmonic morphism from an orientable Einstein 4-manifold M4 to a Riemann surface, or a conformal foliation of M4 by minimal surfaces, determines an (integrable) Hermitian structure with respect to which it is holomorphic. Conversely, any nowhere-Kähler Hermitian structure of an orientable anti-self-dual Einstein 4-manifold arises locally in this way. In the case M4=ℝ4 we show that a Hermitian structure, viewed as a map into S2, is a harmonic morphism; in this case and for S4, [Formula: see text] we determine all (submersive) harmonic morphisms to surfaces locally, and, assuming a non-degeneracy condition on the critical points, globally.

scholarly journals Forbidden sets of planar rational systems of difference equations with common denominator

2014 ◽  
Vol 8 (1) ◽  
pp. 16-32 ◽  
Author(s):  
Ignacio Bajo
Keyword(s):  
Fixed Point ◽  
Finite Number ◽  
Difference Equations ◽  
Spectral Properties ◽  
Common Denominator ◽  
Invariant Line ◽  
First Order ◽  
Rational Difference Equations ◽  
Large Subset ◽  
Associated Matrix

The forbidden sets of systems of first order rational difference equations in the plane in which the denominators are common for all the components of the system is studied. Such forbidden sets are composed of lines which, depending of some spectral properties of an associated matrix, can either be a finite number or lines or an infinity of lines converging to either an invariant line or to a finite number of lines itersecting in a fixed point or else it can be dense in a large subset of R2.

scholarly journals A note on polyhedral cones

1976 ◽  
Vol 22 (4) ◽  
pp. 456-461 ◽  
Author(s):  
Bit-Shun Tam
Keyword(s):  
Finite Number ◽  
Short Note ◽  
Dual Cone ◽  
Polyhedral Cones ◽  
Closed Cone

AbstractIn this short note, two results on a solid, pointed, closed cone C in Rn will be given: first, C is polyhedral iff it has a finite number of maximal faces; second, for any face F of C, C* ∩ F┴ is a face of its dual cone C* of dimension n – dim F.

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