/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. The theory of sets of finite perimeter provides a simple and effective framework. Topics covered include existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems, starting from the basics about Hausdorff measures in Euclidean spaces and ending with complete proofs of the regularity of area-minimizing hypersurfaces up to singular sets of codimension 8. Explanatory pictures, detailed proofs, exercises and remarks providing heuristic motivation and summarizing difficult arguments make this graduate-level textbook suitable for self-study and also a useful reference for researchers. Readers require only undergraduate analysis and basic measure theory.

/pdf/sets-of-finite-perimeter-and-geometric-variational-problems-20hqjijh88.pdf

Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory

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Metric Spaces Of Non Positive Curvature

On the Heegaard Floer homology of branched double-covers

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

3-Manifold Groups

Thurston’s Ending Lamination Conjecture states that a hyperbolic 3manifold N with nitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups; the general case when N has incompressible ends relative to its cusps follows readily. The main ingredient is a uniformly bilipschitz model for the quotient of H 3 by a Kleinian surface group.

/pdf/the-classification-of-kleinian-surface-groups-ii-the-ending-ev79rhttiu.pdf

The classification of kleinian surface groups, ii: the ending lamination conjecture

Thurston's Ending Lamination Conjecture states that a hyperbolic 3-manifold N with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups; the general case when N has incompressible ends relative to its cusps follows readily. The main ingredient is the establishment of a uniformly bilipschitz model for a Kleinian surface group. The first half of the proof appeared in math.GT/0302208, and a subsequent paper will establish the Ending Lamination Conjecture in general.

The classification of Kleinian surface groups, II: The Ending Lamination Conjecture

We present a coarse interpretation of the Weil-Petersson distance dWP(X,Y ) between two finite area hyperbolic Riemann surfaces X and Y using a graph of pants decompositions introduced by Hatcher and Thurston. The combinatorics of the pants graph reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold Q(X,Y ) with X and Y in its conformal boundary is comparable to the Weil-Petersson distance dWP(X,Y ). In applications, we relate the Weil-Petersson distance to the Hausdorff dimen- sion of the limit set and the lowest eigenvalue of the Laplacian for Q(X,Y ), and give a new finiteness criterion for geometric limits. Mathematics Department, University of Chicago, 5734 S. University Ave., Chicago, IL 60637 E-mail address: brock@math.uchicago.edu

/pdf/the-weil-petersson-metric-and-volumes-of-3-dimensional-495edclvob.pdf

The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores

We introduce a coarse combinatorial description of the Weil-Petersson distance d_WP(X,Y) between two finite area hyperbolic Riemann surfaces X and Y. The combinatorics reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold Q(X,Y) with X and Y in its boundary is comparable to the Weil-Petersson distance d_WP(X,Y). Applications include a connection of the Weil-Petersson distance with the Hausdorff dimension of the limit set and the lowest eigenvalue of the Laplacian as well as a new finiteness criterion for geometric limits.

Of course the circle is the least-perimeter way to enclose a region of prescribed area in the plane. This paper proves that a certain standard «double bubble» is the least-perimeter way to enclose and separate two regions of prescribed areas. The solution for three regions remains conjectural

https://msp.org/pjm/1993/159-1/pjm-v159-n1-p03-p.pdf

Jeffrey Brock

Papers

The classification of kleinian surface groups, ii: the ending lamination conjecture

The classification of Kleinian surface groups, II: The Ending Lamination Conjecture

The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores

The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores

The standard double soap bubble in R2 uniquely minimizes perimeter