A New Parabolic Flow in Kahler Manifolds

By Xiuxiong Chen. 2000

82 citations recorded by the source corpus.

This is a follow-up work of my earlier paper [8]. In [8], we study the lower bound of the K energy on the Kahler manifold when the first Chern class is negative. This is an important problem in Kahler geometry since the existence of lower bound of the K energy is the pre-condition for the existence of constant scalar curvature metric problem (cf. [2] and [9]). According to a decomposition formula in [8]2, the problem is reduced to the problem of solving the existence of critical metrics of a new functional J introduced both in our paper [8] and Donaldson’s work [11]. For convenience, we include its definition below. The existence problem is completed solved in Kahler surface. However, the existence problem in general dimension is still open. In this paper, we try to understand the general existence problem in Kahler manifold via flow method. Let (V n, ω0) be a Kahler manifold and ω0 be any Kahler form in V. Consider the space of Kahler potentials

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fcc4d18bec8433567d553da0de73ce75d3207e6d

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Citations in the corpus

The J-flow and stability (2013, methodology): “According to Chen [2] it is enough to show that [cω − α] is a Kähler class.”
Nakai-Moishezon criterions for complex Hessian equations (2020, background): “Both the J-equation and the J-flow are extensively studied in [3, 4, 22, 23, 19, 13, 20, 16, 7, 21, 12, 6].”
On the convergence and singularities of the J‐Flow with applications to the Mabuchi energy (2004, background): “Bounds on all the derivatives of φ and the convergence to a critical metric then follow by the same arguments as in [ Ch3 , We1, We2].”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, methodology): “This would be useful in providing a unified approach to parabolic Hölder estimates for parabolic flows in complex geometry, including theKähler–Ricci flow, the Chern–Ricci flow, the J-flow and their generalizations (see [8,9,11,14,23,29,46,59,68] for example).”
A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics (2010, background): “Hence the Hessian of Kθ (defined wrt the metric g above) is a semipositive Hermitian operator which implies that the corresponding flow decreases the length of any initial curve and is hence distance decreasing (compare the proofs in [21] and [27]).”
On a class of fully nonlinear elliptic equations on Hermitian manifolds (2013, methodology): “It was studied by Chen [5], Weinkove [32,33], Song and Weinkove [28] using parabolic methods.”
Tian's properness conjectures and Finsler geometry of the space of Kahler metrics (2015, background): “To cite a few papers from a rapidly growing literature, we mention [16, 43, 52, 19, 53, 46, 69, 68] and references therein.”