Non-linear Analysis in Riemannian Geometry

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AbstractAward: DMS-0306495Principal Investigator: Jose F. EscobarProfessor Escobar plans to continue his research in linear andnon-linear partial differential equations and its applications toRiemmannian Geometry. He will continue his study of solutions toYamabe equations on compact Riemannian manifolds. These aresemilinear elliptic equations and in the case that the manifoldhas a boundary they satisfy a non-linear boundary condition.Applications of the existence theory for these equations areproposed to solve other problems in differential geometry andrelativity. He also proposes to study Hamilton's Ricci flow ontwo and three dimensional manifols with boundary. In the twodimensional case this is a scalar non-linear evolution equationsatisfying a non-linear boundary condition, while in the threedimensional case is a system of nonlinear equations withnonlinear boundary conditions. He will continue the study of therelation between the geometry of the manifold and the eigenvaluesfor the Steklov problem, which is the study of harmonic functionswhose normal derivative is proportional to the function.The geometric objects that will be studied are the so-calledRiemannian manifolds. These are spaces endowed with analyticalstructures, like the metric which provide us with a way tomeasure lengths and angles. It is natural to study deformationsof these structures to realize what properties in the spaceremain stable under such perturbations. The description of allthese deformations is usually governed by differential equations.The curvature tensor of a Riemmannian manifold ( a measure forthe "non-euclideanness" of a Riemannian space) usually makes suchequations non-linear, although as in physics, most of them are ofvariational nature. From the earliest days, conformal changes ofmetric ( multiplication of the metric by a positive function)have played an importntant role in surface theory. The equationsproposed appear in the problem of conformal deformation of aRiemannian metric and in relativity. The Steklov problem appearsin mathematical physics, conformal geometry, spinor geometry,minimal surfaces, partial differential equations and harmonicanalysis. Geometry as well as topology involves the study ofproperties of the space. But whereas geometry focuses onproperties of space that involves size, shape and measurement,topology concerns itself with the less tangible properties ofrelative position and connectedness. In recent years Hamilton'sRicci flow has become a fundamental tool in the study of geometryand toplogy of manifolds.