New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in Rn
New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in Rn

Author: Antonio Alarcón

Publisher: American Mathematical Soc.

Published: 2020-05-13

Total Pages: 90

ISBN-13: 1470441616

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All the new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables the authors to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, they construct proper non-orientable conformal minimal surfaces in Rn with any given conformal structure, complete non-orientable minimal surfaces in Rn with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits n hyperplanes of CPn−1 in general position, complete non-orientable minimal surfaces bounded by Jordan curves, and complete proper non-orientable minimal surfaces normalized by bordered surfaces in p-convex domains of Rn.

Minimal Surfaces from a Complex Analytic Viewpoint
Minimal Surfaces from a Complex Analytic Viewpoint

Author: Antonio Alarcón

Publisher: Springer Nature

Published: 2021-03-10

Total Pages: 430

ISBN-13: 3030690563

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This monograph offers the first systematic treatment of the theory of minimal surfaces in Euclidean spaces by complex analytic methods, many of which have been developed in recent decades as part of the theory of Oka manifolds (the h-principle in complex analysis). It places particular emphasis on the study of the global theory of minimal surfaces with a given complex structure. Advanced methods of holomorphic approximation, interpolation, and homotopy classification of manifold-valued maps, along with elements of convex integration theory, are implemented for the first time in the theory of minimal surfaces. The text also presents newly developed methods for constructing minimal surfaces in minimally convex domains of Rn, based on the Riemann–Hilbert boundary value problem adapted to minimal surfaces and holomorphic null curves. These methods also provide major advances in the classical Calabi–Yau problem, yielding in particular minimal surfaces with the conformal structure of any given bordered Riemann surface. Offering new directions in the field and several challenging open problems, the primary audience of the book are researchers (including postdocs and PhD students) in differential geometry and complex analysis. Although not primarily intended as a textbook, two introductory chapters surveying background material and the classical theory of minimal surfaces also make it suitable for preparing Masters or PhD level courses.

New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in Mathbb{R}^{n}
New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in Mathbb{R}^{n}

Author: Antonio Alarcón

Publisher:

Published: 2020

Total Pages: 77

ISBN-13: 9781470458126

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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 the authors develop essentially from the first principles, enable them to prove that the space of conformal minimal immersions of a given bordered non-orientable surface to \mathbb{R}^n is a real analytic Banach manifold, obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces, and show general position theorems for non-orientable co.

Global Smooth Solutions for the Inviscid SQG Equation
Global Smooth Solutions for the Inviscid SQG Equation

Author: Angel Castro

Publisher: American Mathematical Soc.

Published: 2020-09-28

Total Pages: 89

ISBN-13: 1470442140

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In this paper, the authors show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation.

Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case
Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case

Author: Jacob Bedrossian

Publisher: American Mathematical Soc.

Published: 2020-09-28

Total Pages: 154

ISBN-13: 1470442175

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The authors study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. They prove that for sufficiently regular initial data of size $epsilon leq c_0mathbf {Re}^-1$ for some universal $c_0 > 0$, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t rightarrow infty $. For times $t gtrsim mathbf {Re}^1/3$, the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of ``2.5 dimensional'' streamwise-independent solutions referred to as streaks.

The Riesz Transform of Codimension Smaller Than One and the Wolff Energy
The Riesz Transform of Codimension Smaller Than One and the Wolff Energy

Author: Benjamin Jaye

Publisher: American Mathematical Soc.

Published: 2020-09-28

Total Pages: 97

ISBN-13: 1470442132

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Fix $dgeq 2$, and $sin (d-1,d)$. The authors characterize the non-negative locally finite non-atomic Borel measures $mu $ in $mathbb R^d$ for which the associated $s$-Riesz transform is bounded in $L^2(mu )$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-Delta )^alpha /2$, $alpha in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.

Conformal Graph Directed Markov Systems on Carnot Groups
Conformal Graph Directed Markov Systems on Carnot Groups

Author: Vasileios Chousionis

Publisher: American Mathematical Soc.

Published: 2020-09-28

Total Pages: 153

ISBN-13: 1470442159

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The authors develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, they develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen's parameter. They illustrate their results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces.

Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields

Author: Lisa Berger

Publisher: American Mathematical Soc.

Published: 2020-09-28

Total Pages: 131

ISBN-13: 1470442191

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The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $mathbb F_p(t)$, when $p$ is prime and $rge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $mathbb F_q(t^1/d)$.

Filtrations and Buildings
Filtrations and Buildings

Author: Christophe Cornut

Publisher: American Mathematical Soc.

Published: 2020-09-28

Total Pages: 150

ISBN-13: 1470442213

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The author constructs and studies a scheme theoretical version of the Tits vectorial building, relates it to filtrations on fiber functors, and uses them to clarify various constructions pertaining to affine Bruhat-Tits buildings, for which he also provides a Tannakian description.

Differential Function Spectra, the Differential Becker-Gottlieb Transfer, and Applications to Differential Algebraic K-Theory
Differential Function Spectra, the Differential Becker-Gottlieb Transfer, and Applications to Differential Algebraic K-Theory

Author: Ulrich Bunke

Publisher: American Mathematical Soc.

Published: 2021-06-21

Total Pages: 177

ISBN-13: 1470446855

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We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the foundational aspects of differential cohomology, including differential function spectra and the differential Becker-Gottlieb transfer. We then state a transfer index conjecture about the equality of the Becker-Gottlieb transfer and the analytic transfer defined by Lott. In support of this conjecture, we derive some non-trivial consequences which are provable by independent means.