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4. Geometry of Moduli Spaces

    1. Diameter of Moduli of Kahler-Einstein Metrics of General Type


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      Problem 4.1.

      [Jacob Sturm] Does the moduli space of Kahler-Einstein manifolds of general type equipped with the Weil-Petersen metric have bounded diameter? It is known that local potentials are continuous up to the boundary. Also, is the $p$-Nash entropy of the Weil-Petersen metric bounded (with respect to the Euclidean reference metric)?

      This is known to be true for curves. What about with respect to the Hodge metric?

        • Properness of Harmonic Map Energy


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          Problem 4.2.

          [Yingying Zhang] Given a Teichmuller space of Kahler-Einstein manifolds (this is the universal cover of the moduli space) of general type, the energy functional for maps into a Riemannian manifold with Hermitian nonpositive curvature is plurisubharmonic. Is this functional also proper? This is related to degenerations of Kahler-Einstein manifolds of general type, and to understanding the boundary of the Teichmuller space.

          This is known for higher-genus curves [Schoen-Yau].

              Cite this as: AimPL: PDE methods in complex geometry, available at http://aimpl.org/pdecomplexgeom.