https://www.math.snu.ac.kr/board/files/attach/images/701/ff97c54e6e21a4ae39315f9a12b27314.png
Extra Form
Lecturer Marshall Slemrod
Dept. Univ. of Wisconsin
date Oct 14, 2010
A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold M2 which can be realized as isometric immersions into R3. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptichyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in R3. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in R3. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a C1,1 isometric immersion of the two-dimensional manifold in R3 satisfying our prescribed initial conditions. To achieve this, we introduce a vanishing viscosity method depending on the features of initial value problems for isometric immersions and present a technique to make the apriori estimates including the L∞ control and H?1?compactness for the viscous approximate solutions. This yields the weak convergence of the vanishing viscosity approximate solutions and the weak continuity of the Gauss-Codazzi system for the approximate solutions, hence the existence of an isometric immersion of the manifold into R3 satisfying our initial conditions.
Atachment
Attachment '1'
List of Articles
Category Subject Dept. Lecturer
Math Colloquia Recent progress on the Brascamp-Lieb inequality and applications file Saitama University Neal Bez
Math Colloquia Connes's Embedding Conjecture and its equivalent file RIMS Narutaka Ozawa
Math Colloquia Class field theory for 3-dimensional foliated dynamical systems file Kyushu University Morishita Masanori
Math Colloquia Unprojection file University of Warwick / 서강대 Miles Reid
Math Colloquia A new view of Fokker-Planck equations in finite and Infinite dimensional spaces file Bielefeld Univ./Purdue Univ. Michael Roeckner
Math Colloquia Convex and non-convex optimization methods in image processing file Hong Kong Baptist University Michael Ng
Math Colloquia Sheaf quantization of Hamiltonian isotopies and non-displacability problems file Kyoto Univ./서울대학교 Masaki Kashiwara
Math Colloquia Codimension Three Conjecture file 교토대학교/서울대학교 Masaki Kashiwara
Math Colloquia Categorical representation theory, Categorification and Khovanov-Lauda-Rouquier algebras file Kyoto University/서울대학교 Masaki Kashiwara
Math Colloquia Riemann-Hilbert correspondence for irregular holonomic D-modules file 서울대학교/RIMS Masaki Kashiwara
Math Colloquia The classification of fusion categories and operator algebras file Kyoto University Masaki Izumi
Math Colloquia Conservation laws and differential geometry file Univ. of Wisconsin Marshall Slemrod
Math Colloquia Quasi-homomorphisms into non-commutative groups file Kyoto Univ. Koji Fujiwara
Math Colloquia Number theoretic results in a family file Univ. of Toronto / KIAS Kim, Henry
Math Colloquia A-infinity functor and topological field theory file Simons Center for Geometry and Physics Kenji Fukaya
Math Colloquia Deformation spaces of Kleinian groups and beyond file Osaka University Kenichi Ohshika
Math Colloquia Analytic torsion and mirror symmetry file Kyoto University Ken-ichi Yoshikawa
Math Colloquia Faithful representations of Chevalley groups over quotient rings of non-Archimedean local fields file Univ. Bremen Keivan Mallahi-Karai
Math Colloquia Conformal field theory and noncommutative geometry file 동경대학교 Kawahigashi
BK21 FOUR Rookies Pitch 2022-1 Rookies Pitch: Harmonic Analysis (Kalachand Shuin) file BK21 Kalachand Shuin
Board Pagination Prev 1 ... 4 5 6 7 8 9 10 11 12 13 14 15 Next
/ 15