As on arXiv
Abstract
Let be a Fano manifold which admits a Kähler-Ricci soliton, we analyze the behavior of the Kähler-Ricci flow near this soliton as we deform the complex structure . First, we will establish an inequality of Lojasiewicz's type for Perelman's entropy along the Kähler-Ricci flow. Then we prove the convergence of Kähler-Ricci flow when the complex structure associated to the initial value lies in the kernel or negative part of the second variation operator of Perelman's entropy. As applications, we solve the Yau-Tian-Donaldson conjecture for the existence of Kähler-Ricci solitons in the moduli space of complex structures near , and we show that the kernel corresponds to the local moduli space of Fano manifolds which are modified -semistable. We also prove an uniqueness theorem for Kähler-Ricci solitons.
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.
-
Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds
Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Same authors Gang Tian, Xiaohua Zhu · Shared terms “Fano”, “Lie”
-
Laplace comparison on Kähler Ricci flow and convergence
Topics Kähler–Ricci flow, Kähler geometry · Same authors Gang Tian, Xiaohua Zhu · Shared terms “Fano manifold”, “Tian”
-
Kähler-Ricci flow on -spherical Fano manifolds
Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu · Shared terms “Fano manifold”, “Lie”
-
Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds
Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler–Ricci flow · Same author Xiaohua Zhu · Shared terms “Fano manifold”
-
Lower bound of modified -energy on a Fano manifold with degeneration for Kähler-Ricci solitons
Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Same author Liang Zhang · Shared terms “Fano manifold”, “modified”
-
A Note On Kähler-Ricci Flow on Fano Threefolds
Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Same author Gang Tian · Shared terms “Fano”
-
A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds
Topics Kähler–Ricci flow, Kähler geometry · Same author Gang Tian · Shared terms “Fano manifold”, “Perelman”
-
Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein
Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “Fano manifold”
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
- 6A.12Entropy, Information and Diffusion
Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.
- 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.
- 6B.16Eigenvalue Problems and Perelman’s λ
Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.