WorldCat Identities

Brendle, Simon 1981-

Overview
Works: 17 works in 35 publications in 3 languages and 414 library holdings
Roles: Author, Contributor, Thesis advisor, Author of introduction
Publication Timeline
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Most widely held works by Simon Brendle
Ricci flow and the sphere theorem by Simon Brendle( Book )

10 editions published in 2010 in English and held by 214 WorldCat member libraries worldwide

"In 1982, R. Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds. This evolution equation is known as the Ricci flow, and it has since been used widely and with great success, most notably in Perelman's solution of the Poincare conjecture. Furthermore, various convergence theorems have been established. This book provides a concise introduction to the subject as well as a comprehensive account of the convergence theory for the Ricci flow. The proofs rely mostly on maximum principle arguments. Special emphasis is placed on preserved curvature conditions, such as positive isotropic curvature. One of the major consequences of this theory is the Differentiable Sphere Theorem: a compact Riemannian manifold, whose sectional curvatures all lie in the interval (1,4], is diffeomorphic to a spherical space form. This question has a long history, dating back to a seminal paper by H.E. Rauch in 1951, and it was resolved in 2007 by the author and Richard Schoen."--Publisher's description
Ricci flow and the sphere theorem by Simon Brendle( )

3 editions published in 2010 in English and held by 65 WorldCat member libraries worldwide

"In 1982, R. Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds. This evolution equation is known as the Ricci flow, and it has since been used widely and with great success, most notably in Perelman's solution of the Poincare conjecture. Furthermore, various convergence theorems have been established. This book provides a concise introduction to the subject as well as a comprehensive account of the convergence theory for the Ricci flow. The proofs rely mostly on maximum principle arguments. Special emphasis is placed on preserved curvature conditions, such as positive isotropic curvature. One of the major consequences of this theory is the Differentiable Sphere Theorem: a compact Riemannian manifold, whose sectional curvatures all lie in the interval (1,4], is diffeomorphic to a spherical space form. This question has a long history, dating back to a seminal paper by H.E. Rauch in 1951, and it was resolved in 2007 by the author and Richard Schoen."--Publisher's description
Geometry and topology : lectures given at the Geometry and Topology conferences at Harvard University in 2011 and at Lehigh University in 2012( Book )

1 edition published in 2013 in English and held by 59 WorldCat member libraries worldwide

This volume includes papers presented by several speakers at the Geometry and Topology conferences at Harvard University in 2011 and at Lehigh University in 2012. Included are works by Simon Brendle, on the Lagrangian minimal surface equation and related problems; by Sergio Cecotti and Cumrun Vafa, concerning classification of complete N=2 supersymmetric theories in four dimensions; by F. Reese Harvey and H. Blaine Lawson Jr., on existence, uniqueness, and removable singularities for non-linear PDEs in geometry; by János Kollár, concerning links of complex analytic singularities; by Claude LeBrun, on Calabi energies of extremal toric surfaces; by Mu-Tao Wang, concerning mean curvature flows and isotopy problems; and by Steve Zelditch, on eigenfunctions and nodal sets
One-parameter semigroups for linear evolution equations by Klaus-Jochen Engel( Book )

5 editions published between 2000 and 2008 in English and Italian and held by 41 WorldCat member libraries worldwide

"This book gives an up-to-date account of the theory of strongly continuous one-parameter semigroups of linear operators. It includes a systematic discussion of the spectral theory and the long-term behavior of such semigroups. A special feature of the text is an unusually wide range of applications, to ordinary and partial differential operators, delay and Volterra equations and to control theory, et cetera, and an emphasis on philosophical motivation and the historical background."--BOOK JACKET. "The book is written for students, but should also be of value for researchers interested in this field."--Jacket
Krümmungsflüsse auf Mannigfaltigkeiten mit Rand by Simon Brendle( Book )

3 editions published in 2001 in German and held by 12 WorldCat member libraries worldwide

Ramification theory for varieties over a local field( Book )

2 editions published in 2013 in English and held by 7 WorldCat member libraries worldwide

One-parameter semigroups for linear evolution equations by Klaus-Jochen Engel( Book )

1 edition published in 1999 in English and held by 6 WorldCat member libraries worldwide

The Ninth Takagi lectures 4 June 2011, Kyoto by Carlos E Kenig( Book )

1 edition published in 2011 in English and held by 1 WorldCat member library worldwide

New results on the singularity analysis of the Kaehler-Ricci flow by Tsz Ho Fong( )

1 edition published in 2012 in English and held by 1 WorldCat member library worldwide

In this thesis, we study the singularity development of the Kaehler-Ricci flow on holomorphic fibrations, and classify the singularity models of some classes of fibrations using parabolic rescaling. We first study the collapsing behavior of Calabi-Yau fibrations under the Kaehler-Ricci flow. A compact Kaehler manifold with semi-ample canonical line bundle admits a fibration of Calabi-Yau manifolds with possibly singular fibers. The convergence behavior for this class of manifolds under the Kaehler-Ricci flow was first studied by Song and Tian who establish the metric convergence in the sense of currents. In this thesis, we obtain the optimal collapsing rate of the nonsingular Calabi-Yau fibers, thus improving Song and Tian's work in analytic and geometric aspects. Secondly, we focus on a specific type of holomorphic fibration, namely the CP^1-bundles over Kaehler-Einstein manifolds. Fiber collapsing in the sense of Gromov-Hausdorff convergence was shown to occur in this case by Song, Szekelyhidi and Weinkove. We study the finite-time singularities for these manifolds using parabolic rescaling and dilation procedures adapted from Hamilton and Perelman in their works of the Ricci flow on 3-manifolds. We prove that when the flow metric has cohomogeneity-1 symmetry the collapsing occurs as a Type I singularity and we show that the singularity is modelled by C^n X CP^1
Elliptic and parabolic problems in conformal geometry by Simon Brendle( )

1 edition published in 2006 in English and held by 1 WorldCat member library worldwide

Wave equations on asymptotically de Sitter spaces by Dean Russell Baskin( )

1 edition published in 2010 in English and held by 1 WorldCat member library worldwide

Asymptotically de Sitter spaces are Lorentzian manifolds modeled on the de Sitter space of general relativity. In this dissertation, we construct the forward fundamental solution for the wave and Klein-Gordon equations on asymptotically de Sitter spaces. We adapt classes of conormal and paired Lagrangian distributions to this setting and show that the lift of the kernel of the forward fundamental solution to a blown-up space is a sum of distributions in these classes. We use the structure of the kernel of the fundamental solution to study its mapping properties. We show that Strichartz estimates with loss hold for the positive mass Klein-Gordon equation on asymptotically de Sitter spaces. When the mass parameter is the conformal value, Strichartz estimates hold without loss. As an application of these estimates, we prove a small-data global existence result for a defocusing Klein-Gordon equation
Decay of rhenish tuff in Dutch monuments( )

1 edition published in 2004 in English and held by 1 WorldCat member library worldwide

On the free-boundary mean curvature flow by Nicholas Edelen( )

1 edition published in 2016 in English and held by 1 WorldCat member library worldwide

We investigate the free-boundary mean curvature flow. This is an evolution of surfaces by ``steepest descent for area, '' while preserving the Neumann-type condition that all the surfaces meet some fixed barrier orthogonally. For example, a bubble in a sink. We first prove the Huisken-Sinestrari convexity estimates for free-boundary mean curvature flow, which classifies ``type-II'' singularities. We then develop the notion of weak free-boundary mean curvature flow, extending Brakke's original definition, and proving a local regularity theorem. We also prove a geometric eigenvalue gap estimate, extending results of Ashbaugh-Benguria and Benguria-Linde
Regularity theory for the symmetric minimal surface equation by Kaveh Fouladgar( )

1 edition published in 2012 in English and held by 1 WorldCat member library worldwide

We consider the solutions of the symmetric minimal surface equation (SME), the geometric significance of which is that such solutions have symmetric graphs, which are minimal hypersurfaces. The SME has been previously studied by Dierkes, Huisken, and Simon. Using the uniform Holder continuity estimate for positive bounded solutions proved by Simon, we are able to define a singular solution of the SME as local uniform limits of sequences of positive solutions of the SME. Next, we study the existence of singular solutions, beginning with the observation that, by standard ODE theory, one-dimensional singular solutions do not exist. However we will show, via a Leray-Schauder argument, that in all dimensions greater than one there is rich class of singular solutions. Having established an existence theory for the SME, we focus on studying the regularity and singularity properties of singular solutions of the SME. More or less standard quasilinear elliptic theory shows that for any bounded positive solution of the SME there are local gradient bounds, and for positive solutions defined on the closure of a smooth convex domain there is a global gradient bound, depending only on the supremum and infimum of the solution and the domain. For singular solutions (or classes of positive solutions without a uniform positive lower bound), it will be shown that there is a local gradient estimate in a neighborhood of any singular point, depending only on the supremum of the solution and the distance of the singular point and the boundary of the domain. To establish these latter gradient estimates, we will use a modification of Simon's blow up of the Jacobi field argument together with the work of Ilmanen. As a consequence of gradient estimates, we are then able to show, via an application of the Federer dimension reducing argument, that the Hausdorff dimension of the singular set of an n-dimensional SME is at most n-2
Microlocal analysis of lagrangian submanifolds of radial points by Nick Haber( )

1 edition published in 2013 in English and held by 1 WorldCat member library worldwide

Microlocal analysis relies on correspondences between quantum physics and classical physics to give information about certain PDEs -- for instance, linear variable-coefficient PDEs on manifolds. PDEs are interpreted as quantum systems. The corresponding classical systems tell us, for example, function spaces on which problems are solvable or almost solvable, existence and uniqueness results, and the structure of solution operators. Landmark papers of Hörmander and Duistermaat and Hörmander establish key results for the standard calculus of microlocal analysis, which gives a broad framework for dealing with variable-coefficient PDEs on manifolds. Their work is well-suited for dealing with PDEs which, in a generalized sense, are hyperbolic, with corresponding classical dynamics looking like wave propagation of geometric optics. In this thesis, we aim to extend many of their results to situations in which the corresponding classical dynamics are less well-behaved: those with a Lagrangian submanifold of radial points
On the variational methods for minimal submanifolds by Xin Zhou( )

1 edition published in 2013 in English and held by 1 WorldCat member library worldwide

In this thesis, we study two problems concerning variational methods for minimal submanifolds. Specifically we consider min-max theory for minimal surfaces. We construct the "saddle point" type critical points for the area functional, and study the geometrical properties of these. In the first part, we consider the mapping problem for the min-max theory. In particular, we prove an existence theorem for min-max minimal surfaces of arbitrary genus g ≥ 2 by variational methods. We show that the min-max critical value for the area functional can be achieved by the bubbling limit of branched minimal surfaces with nodes of genus g together with possibly finitely many branched minimal spheres. We also prove a strong convergence theorem similar to the classical mountain pass lemma. In the second part, we consider the geometric measure theory approach to the min-max theory. We study the shape of the min-max minimal hypersurface constructed by Almgren-Pitts corresponding to the fundamental class of a Riemannian manifold (M, g) of dimension n + 1 with positive Ricci curvature and 2 ≤ n ≤ 6. We characterize the Morse index, area and multiplicity of this min-max hypersurface. Precisely, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces
The geometry of asymptotically hyperbolic manifolds by Otis Chodosh( )

1 edition published in 2015 in English and held by 1 WorldCat member library worldwide

We discuss the large-scale geometry of asymptotically hyperbolic manifolds. Asymptotically hyperbolic manifolds arise naturally in general relativity. However, several fundamental questions about them remain unresolved, including the asymptotically hyperbolic Penrose inequality and the static uniqueness of the Schwarzschild-anti-de Sitter metric. The main contributions of this thesis are twofold: Firstly, we introduce a new notion of renormalized volume for asymptotically hyperbolic manifolds and prove a sharp Penrose-type inequality where mass is replaced by renormalized volume. Secondly, we use the notion of renormalized volume to study isoperimetric regions in asymptotically hyperbolic manifolds. We prove that for initial data sets that are Schwarzschild-anti-de Sitter at infinity and satisfy appropriate scalar curvature lower bounds, sufficiently large coordinate spheres are uniquely isoperimetric. This is relevant in the context of Bray's isoperimetric approach to the Penrose inequality. From a geometric viewpoint, our results show that the large-scale geometry of asymptotically hyperbolic manifolds significantly differs from the more familiar asymptotically flat setting. The renormalized volume is a very different quantity from the ``mass, '' and our results suggest that it is a stronger quantity. As a consequence of this, we uncover a link between scalar curvature and the behavior of large isoperimetric regions, which is not present in the asymptotically flat setting. Additionally, we discuss isoperimetric regions in warped products and consequences for the renormalized volume of a more general class of metrics. Finally, we study rotational symmetry of expanding Ricci solitons, a problem that is formally similar to the static uniqueness question with negative cosmological constant
 
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WorldCat IdentitiesRelated Identities
Ricci flow and the sphere theorem Ricci flow and the sphere theorem
Covers
Ricci flow and the sphere theoremOne-parameter semigroups for linear evolution equationsOne-parameter semigroups for linear evolution equations
Alternative Names
Brendle, S.

Simon Brendle deutscher Mathematiker

Simon Brendle Duits wiskundige

Simon Brendle German mathematician

Simon Brendle matemàtic alemany

Simon Brendle matemático alemán

Simon Brendle mathématicien allemand

Simon Brendle tysk matematikar

Simon Brendle tysk matematiker

西蒙·布倫德

西蒙·布伦德勒

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