As on arXiv
Abstract
Extending a recent theory developed on compact Kähler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Ampère equations on compact Hermitian manifolds. Under natural assumptions on the Cauchy boundary data, we show that the pluripotential solution is semi-concave in time and continuous in space and that such a solution is unique. We also establish a partial regularity of such solutions under some extra assumptions of the densities and apply it to prove the existence and uniqueness of the weak Chern-Ricci flow on complex compact varieties with log terminal singularities.
Comments: 26 pages. arXiv admin note: text overlap with arXiv:1810.02121 by other authors
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Singularities of the Chern-Ricci flow
Topics Analytic minimal model program, Generalized & G₂ flows, Kähler geometry · Same author Quang-Tuan Dang · Shared terms “Chern Ricci”, “complex Monge”, “degenerate parabolic”
-
Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds
Topics Generalized & G₂ flows, Kähler geometry, Stability & uniqueness · Same author Quang-Tuan Dang · Shared terms “compact Hermitian”, “Hermitian manifolds”, “Ampère equation”
-
Hermitian null loci
Topics Analytic minimal model program, Generalized & G₂ flows, Singularities · Same author Quang-Tuan Dang · Shared terms “Chern”, “Hermitian”, “developed”
-
The Chern-Ricci flow
Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “Hermitian”
-
Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces
Topics Analytic minimal model program, Kähler geometry, Singularities · Shared terms “complex Monge”, “Ampère equation”, “Monge Ampère”
-
Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow
Topics Generalized & G₂ flows, Kähler geometry, Stability & uniqueness · Shared terms “Chern Ricci”, “Hermitian”
-
The behavior of the second Ricci flow on complex parallelizable manifolds
Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “compact Hermitian”, “Hermitian manifolds”
-
Convergence of the Chern-Ricci flow on complex minimal surfaces of general type
Topics Analytic minimal model program, Generalized & G₂ flows, Kähler geometry · Shared terms “Chern Ricci”, “Hermitian”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.4Blow-Ups, Blow-Downs and Splitting at Infinity
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
- 11B.6Rotationally Symmetric Flows and the Standard Solution
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.