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Moduli of Continuity, Gauss Curvature Flow and Ricci Solitons


This thesis is a summary of the work accomplished by the author and his coauthors in geometric analysis during his Ph.D. studies. It consists of four parts.

The first and the second parts are the estimates of modulus of continuity for viscosity solutions of nonlinear partial differential equations in domains in Euclidean spaces and on manifolds. The main results generalize B. Andrews and J. Clutterbuck's modulus of continuity estimates for smooth solutions to viscosity solutions. The main ingredients of the proofs are the parabolic maximum principle for semicontinuous functions, its generalized version on manifolds, and the multi-point estimates method.

The third part studies asymptotic behavior of nonparametric hypersurfaces of dimension $n$

moving by $\a $ powers of its Gaussian Curvature with $\a >1/n$.

Our work generalizes the results for $\a =1$ obtained by V. Oliker to all $\a > 1/n$.

Although we are using similar ideas, the proof is quite technical.

In the fourth part, we study classification of shrinking gradient Ricci solitons.

Our main result asserts that any four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature

is either a quotient of $\mathbb{S}^4$ or a quotient of $\mathbb{S}^3 \times \R $.

This gives a clean classification result removing the earlier additional assumptions in by L. Ni and N. Wallach.

%An immediate corollary is that any four-dimensional gradient shrinking soliton with positive curvature operator must be isometric to $S^4$.

This also generalizes a result of Perelman on three-dimensional gradient shrinking Ricci solitons to dimension four.

The result has important consequences in studying Ricci flow on four-dimensional manifolds.

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