Affine Geometric Analysis

Project Summary The applicant has defended his Ph.D. thesis in the area of geometric flows and convex geometry under the supervision of Alina Stancu at the Concordia University, Montreal, in 2013. Since Fall 2013 he is staying at the Institut für Diskrete Mathematik und Geometrie, TU Wien, as a post-doctoral researcher. The Institut für Diskrete Mathematik und Geometrie, currently with Monika Ludwig as director and Franz Schuster as deputy director, is internationally renowned as an excellent center for mathematics research in convex geometry. Today the Institute brings together a wide range of researchers in a vibrant and enthusiastic community to do the highest caliber of mathematical research. The research project proposed here is at the interface of M. Ludwig`s, F.E. Schuster`s and M.N. Ivaki`s expertise, namely convex geometry and geometric flows. One of the main interests of M. Ludwig and F.E. Schuster are affine isoperimetric inequalities and applications of the affine geometry of convex bodies to PDEs, therefore, the integration of the applicant in the Institute is very natural and both sides should profit substantially from the project. The planned research for a post-doctoral position in 2014-2016 divides into two areas: convex geometry and geometric flows. The applicant aims at employing tools of geometric flows in convex geometry and vice versa to tackle some of the major open problems of each field, such as Andrews`s conjecture on Gauss curvature flow and Petty`s conjectured projection inequality. One of the axes of the present research plan is inequalities and stability of inequalities. It is described here that one may employ geometric flows and tools from the theory of non-linear parabolic differential equations to prove inequalities and moreover exploit tools such as Harnacks estimate and displacement bounds to obtain the stability of inequalities. It is often the case that an appropriate variation of the geometric quantity involved in an inequality yields a ratio in connection to Minkowski`s mixed volume inequality where sharp stability results are already known. The second axis of the present research plan is to develop new techniques for obtaining regularity of solutions in Euclidean ambient space. It is observed by the applicant and Alina Stancu that some objects of convex geometry such as polar convex bodies can be employed as a useful tool to obtain regularity of solutions to some geometric flows. Therefore, it is natural to investigate more deeply further associated convex bodies and explore whether they will give rise to new techniques. Moreover, the applicant aims at investigating geometric flows in background spaces such as Hyperbolic and Riemannian spaces which are of major interest to experts.

During the project, several new results were obtained. Here I mention the most important ones. New results in the area of convex geometry were obtained by the project leader related to operators that take convex bodies to the corresponding i-projection bodies, their iterations, centroid bodies, and polar bodies. Under certain conditions, it is shown that fixed points of the second mixed projection operator and iterates of the projection and centroid operators are balls or ellipsoids. In particular, partial affirmative answers were provided to a few conjectures in convex geometry. In geometric analysis, new methods and techniques were introduced in collaboration with Dr. Paul Bryan and Dr. Julian Scheuer. One of our achievements was extending a class of differential Harnack inequalities to ambient spaces other than the Euclidean space. Differential Harnack inequalities provide useful information about solutions to the curvature flows, and we believe that our results will serve as a platform for the launching of several new applications. In another direction, in collaboration with Dr. Paul Bryan and Dr. Julian Scheuer we introduced a unified approach to the even, smooth Lp Minkowski problem. Our techniques and approach have recently inspired other experts to consider our approach to treat an important class of problems, namely, the dual Lp Minkowski problem.