##### Department of Mathematics,

University of California San Diego

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### Math 258 - Seminar in Differential Geometry

## Gunhee Cho

#### UCSB

## The lower bound of the integrated Carath Ìeodory-Reiffen metric and Invariant metrics on complete noncompact Kaehler manifolds

##### Abstract:

We seek to gain progress on the following long-standing conjectures in hyperbolic complex geometry: prove that a simply connected complete K Ìˆahler manifold with negatively pinched sectional curvature is biholomorphic to a bounded domain and the Carath Ìeodory-Reiffen metric does not vanish everywhere. As the next development of the important recent results of D. Wu and S.T. Yau in obtaining uniformly equivalence of the base K Ìˆahler metric with the Bergman metric, the Kobayashi-Royden metric, and the complete Ka Ìˆhler-Einstein metric in the conjecture class but missing of the Carath Ìeodory-Reiffen metric, we provide an integrated gradient estimate of the bounded holomorphic function which becomes a quantitative lower bound of the integrated Carath Ìeodory-Reiffen metric. Also, without requiring the negatively pinched holomorphic sectional curvature condition of the Bergman metric, we establish the equivalence of the Bergman metric, the Kobayashi-Royden metric, and the complete Ka Ìˆhler-Einstein metric of negative scalar curvature under a bounded curvature condition of the Bergman metric on an n-dimensional complete noncompact Ka Ìˆhler manifold with some reasonable conditions which also imply non-vanishing Carath Ìedoroy-Reiffen metric. This is a joint work with Kyu-Hwan Lee.

### January 27, 2022

### 11:00 AM

AP&M Room 7321

Zoom ID: 949 1413 1783

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