Understanding how to utilize the properties of Kähler manifolds Part1(Differential Geometry)
- Explicit lower bound of the first eigenvalue of the Laplacian on Kähler manifolds(arXiv)
Author : Benjamin Rutkowski, Shoo Seto
Abstract : We establish an explicit lower bound of the first eigenvalue of the Laplacian on Kähler manifolds based off the comparison results of Li and Wang. The lower bound will depend on the diameter, dimension, holomorphic sectional curvature and orthogonal Ricci curvature.
2.Hermitian structures on a class of quaternionic Kähler manifolds (arXiv)
Author : V. Cortés, A. Saha, D. Thung
Abstract : Any quaternionic Kähler manifold (N¯,gN¯,Q) equipped with a Killing vector field X with nowhere vanishing quaternionic moment map carries an integrable almost complex structure J1 that is a section of the quaternionic structure Q. Using the HK/QK correspondence, we study properties of the almost Hermitian structure (gN¯,J~1) obtained by changing the sign of J1 on the distribution spanned by X and J1X. In particular, we derive necessary and sufficient conditions for its integrability and for it being conformally Kähler. We show that for a large class of quaternionic Kähler manifolds containing the one-loop deformed c-map spaces, the structure J~1 is integrable. We do also show that the integrability of J~1 implies that (gN¯,J~1) is conformally Kähler in dimension four, but leads to an extremely stringent constraint in higher dimensions that we show is never satisfied by a quaternionic Kähler manifold in the image of the one-loop deformed supergravity c-map. In the special case of the one-loop deformation of the quaternionic Kähler symmetric spaces dual to the complex Grassmannians of two-planes we construct a third canonical Hermitian structure (gN¯,J¹). Finally, we give a complete local classification of quaternionic Kähler four-folds for which J~1 is integrable and show that these are either locally symmetric or carry a cohomogeneity 1 isometric action generated by one of the Lie algebras o(2)⋉heis3(R), u(2), or u(1,1).
3.A note on quaternionic Kähler manifolds with ends of finite volume (arXiv)
Author : V. Cortés
Abstract : We prove that complete non-locally symmetric quaternionic Kähler manifolds with an end of finite volume exist in all dimensions 4m≥4.
4. Gradient estimates for Donaldson’s equation on a compact Kähler manifold(arXiv)
Author : Liangdi Zhang
Abstract : We prove a gradient estimate for Donaldson’s equation
ω∧(χ+−1−−−√∂∂¯φ)n−1=eF(χ+−1−−−√∂∂¯φ)n
(and its parabolic analog) on an n-dimensional compact Kähler manifold (M,ω) with another Hermitian metric χ directly from the C2 estimate and Alexandrov-Bakelman-Pucci (ABP) maximum principle.
