scholarly journals Expansions of solutions to extremal metric type equations on blowups of cscK surfaces

2018 ◽  
Vol 55 (2) ◽  
pp. 215-241
Author(s):  
Ved V. Datar
Keyword(s):  
Extremal Metric

scholarly journals On the Denjoy Conjecture

1958 ◽  
Vol 10 ◽  
pp. 627-631 ◽  
Author(s):  
James A. Jenkins
Keyword(s):  
Quasiconformal Mappings ◽  
Regular Functions ◽  
Extremal Metric

In recent years many of the properties of regular functions have been shown to extend to quasiconformal mappings. (The latter term is here understood in the sense defined in (5).) This is particularly true of those results which can be proved by use of the method of the extremal metric. It is rather strange, then, that a result which constitutes one of the first notable applications of this method has not been so extended (at least to the author's knowledge).

On a result of M. Heins

1975 ◽  
Vol 19 (4) ◽  
pp. 371-373 ◽  
Author(s):  
James A. Jenkins
Keyword(s):  
Riemann Surface ◽  
Simple Proof ◽  
Main Part ◽  
Plane Domain ◽  
Extremal Metric ◽  
Positive Genus ◽  
Closed Riemann Surface

Some years ago Heins (1) proved that a Riemann surface which can be conformally imbedded in every closed Riemann surface of a fixed positive genus g is conformally equivalent to a bounded plane domain. In the proof the main effort is required to prove that a surface satisfying this condition is schlichtartig. Heins gave quite a simple proof of the remaining portion (1; Lemma 1). The main part of the proof depended on exhibiting a family of surfaces of genus g such that a surface which could be conformally imbedded in all of them was necessarily schlichtartig. Another proof using a different construction was recently given by Rochberg (2). We will give here a further proof based on the method of the extremal metric and using a further construction which is in some ways more direct than those previously given.

A form of the extremal metric method

1968 ◽  
Vol 19 (1) ◽  
pp. 108-112
Author(s):  
P. M. Tamrazov
Keyword(s):  
Extremal Metric

scholarly journals COMPACT NON-ORIENTABLE SURFACES OF GENUS 5 WITH EXTREMAL METRIC DISCS

2011 ◽  
Vol 54 (2) ◽  
pp. 273-281 ◽  
Author(s):  
GOU NAKAMURA
Keyword(s):  
Hyperbolic Surface ◽  
Extremal Surface ◽  
Extremal Metric ◽  
Orientable Surfaces

AbstractA compact hyperbolic surface of genus g is called an extremal surface if it admits an extremal disc, a disc of the largest radius determined by g. Our problem is to find how many extremal discs are embedded in non-orientable extremal surfaces. It is known that non-orientable extremal surfaces of genus g > 6 contain exactly one extremal disc and that of genus 3 or 4 contain at most two. In the present paper we shall give all the non-orientable extremal surfaces of genus 5, and find the locations of all extremal discs in those surfaces. As a consequence, non-orientable extremal surfaces of genus 5 contain at most two extremal discs.

scholarly journals Optimal systolic inequalities on Finsler Möbius bands

2016 ◽  
Vol 08 (02) ◽  
pp. 349-372 ◽  
Author(s):  
Stéphane Sabourau ◽  
Zeina Yassine
Keyword(s):  
Klein Bottle ◽  
Finsler Metrics ◽  
Möbius Band ◽  
Extremal Metric ◽  
Systolic Inequality

We prove optimal systolic inequalities on Finsler Möbius bands relating the systole and the height of the Möbius band to its Holmes–Thompson volume. We also establish an optimal systolic inequality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric families both on the Möbius band and the Klein bottle are also presented.

scholarly journals Classical Variational Theory of the Cosmological Constant and Its Consistency with Quantum Prescription

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 633 ◽  
Author(s):  
Claudio Cremaschini ◽  
Massimo Tessarotto
Keyword(s):  
Variational Principle ◽  
Cosmological Constant ◽  
Lagrange Multiplier ◽  
Proper Time ◽  
Variational Theory ◽  
Field Equations ◽  
Metric Tensor ◽  
Covariant Quantum ◽  
Hamilton Variational Principle ◽  
Extremal Metric

The manifestly-covariant Hamiltonian structure of classical General Relativity is shown to be associated with a path-integral synchronous Hamilton variational principle for the Einstein field equations. A realization of the same variational principle in both unconstrained and constrained forms is provided. As a consequence, the cosmological constant is found to be identified with a Lagrange multiplier associated with the normalization constraint for the extremal metric tensor. In particular, it is proved that the same Lagrange multiplier identifies a 4-scalar gauge function generally dependent on an invariant proper-time parameter s. Such a result is shown to be consistent with the prediction of the cosmological constant based on the theory of manifestly-covariant quantum gravity.

Geometric Proofs of some Classical Results on Boundary Values for Analytic Functions

1994 ◽  
Vol 37 (2) ◽  
pp. 263-269 ◽  
Author(s):  
Enrique Villamor
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Logarithmic Capacity ◽  
Geometric Proof ◽  
Boundary Values ◽  
Dirichlet Integral ◽  
Capacity Zero ◽  
Extremal Metric ◽  
Nontangential Limits

AbstractIn this note we are going to give a geometric proof, using the method of the extremal metric, of the following result of Beurling. For any analytic function f(z) in the unit disc Δ of the plane with a bounded Dirichlet integral, the set E on the boundary of the unit disc where the nontangential limits of f(z) do not exist has logarithmic capacity zero. Also, using an unpublished result of Beurling, we will prove different results on boundary values for different classes of functions.

scholarly journals A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle

2006 ◽  
Vol 135 (1) ◽  
pp. 181-202 ◽  
Author(s):  
Ahmad El Soufi ◽  
Hector Giacomini ◽  
Mustapha Jazar
Keyword(s):  
Klein Bottle ◽  
Least Eigenvalue ◽  
Extremal Metric
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