Partial Regularity of Harmonic Maps From Alexandrov Spaces

Topological regularity theorems for Alexandrov spaces

Since Gromov gave in [G1], [G2] an abstract definition of Hausdorff distance between two compact metric spaces, the Gromov-Hausdorff convergence theory has played an important role in Riemannian geometry. Usually, Gromov-Hausdorff limits of Riemannian manifolds are almost never Riemannian manifolds. This motivates the study of Alexandrov spaces which are more singular than Riemannian manifolds ...

Regularity of Harmonic Maps

Regularity of Dirac-harmonic maps

For any n-dimensional compact spin Riemannian manifold M with a given spin structure and a spinor bundle ΣM , and any compact Riemannian manifold N , we show an ǫ-regularity theorem for weakly Dirac-harmonic maps (φ, ψ) : M ⊗ΣM → N ⊗ φ∗TN . As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is establi...

Regularity for n-Harmonic Maps

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Harmonic Functions on Alexandrov Spaces and Their Applications

The main result can be stated roughly as follows: Let M be an Alexandrov space, Ω ⊂M an open domain and f : Ω→ R a harmonic function. Then f is Lipschitz on any compact subset of Ω. Using this result I extend proofs of some classical theorems in Riemannian geometry to Alexandrov spaces.