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Geometric Analysis and Partial Differential Equations Seminar
August 8, 2013 @ 10:30 am - 11:30 pm
Dr Glen Wheeler (University of Wollongong)
Gap phenomena for a class of fourth-order geometric differential operators on general surfaces with boundary
Gap phenomena are a kind of geometric rigidity which prevent an associated tensor from being small in an appropriate norm: it is either larger than a universal constant, or identically zero. I will survey the idea behind classical gap theorems, providing a context for my own results (some joint with James McCoy). I aim to explain in detail the main results of my most recent preprint, which are a pair of gap theorems where the associated tensor is the second fundamental form or the trace-free second fundamental form. Key improvements over previous work are the generalisations to high codimension, surfaces with arbitrary topology and boundaries, and more general operators. The class of operators considered include the motivating examples of the Willmore operator, the Surface Diffusion operator (normal Laplacian) and the biharmonic or Chen’s operator (bilaplacian). Each of the gap theorems are new even for these model cases.