Rigidity Results of Lambda-Hypersurfaces

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Johns Hopkins University
This thesis introduces lambda-hypersurfaces. These are n-dimensional hypesrufaces immersed in n+1 Euclidean space that satisfy a specific mean curvature equation. Such hypersurfaces generalize the notion of a self-shrinking soliton of mean curvature flow. They also are stationary solutions to an isoperimetric-type problem on a Gaussian measure. We will motivate the study of lambda-hypersurfaces, then give several rigidity results that can help to classify such surfaces. These results include a stability result (that only stable hypersurfaces, suitably defined, are hyperplanes) with versions that apply to both complete and incomplete hypersufaces. The results also include an eigenvalue and diameter estimate for compact hypersurfaces. Finally, we state a classification result about compact surfaces with small curvature. Advisor: William P. Minicozzi II
Isoperimetric Problem, Gaussiam Measure, Mean Curvature Flow, Stability