Convergence of the parabolic complex Monge-Amp\`ere equation on compact Hermitian manifolds

By M. Gill. 2010

143 citations recorded by the source corpus.

We prove $C^\infty$ convergence for suitably normalized solutions of the parabolic complex Monge-Amp\`ere equation on compact Hermitian manifolds. This provides a parabolic proof of a recent result of Tosatti and Weinkove.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0e3d20ffa21bd2a05b54129af076c953ce2bce95

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The Monge-Ampère equation for (n − 1)-plurisubharmonic functions on a compact Kähler manifold (2013, background): “coincides with the Chern-Ricci flow (see [22, 52, 53, 46, 54]) and the KählerRicci flow if the initial metric ω0 is Kähler.”
Non-Kähler Calabi-Yau manifolds (2014, methodology): “Furthermore, as shown by Gill [34], on such manifolds the Chern-Ricci flow [73, 74, 77] deforms any given Hermitian metric to a Chern-Ricci flat one.”
$$C^{2,\alpha }$$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).”
Fully non-linear parabolic equations on compact Hermitian manifolds (2017, methodology): “For this, we adapt the arguments of Cao [2] and especially Gill [17] based on the Harnack inequality.”
Ricci Curvatures on Hermitian manifolds (2014, methodology): “[45, 46, 47], [39], [24, 51, 52, 55, 25, 26]) by using the well-studied Hamilton’s Ricci flow.”
The Chern–Ricci flow on complex surfaces (2012, methodology): “Recall that Gill [21] showed, in the setting of vanishing first Bott-Chern class, the Chern-Ricci flow ω(t) starting at any Hermitian metric ω0 converges in C∞ to a Chern-Ricci flat metric ω∞.”
On the evolution of a Hermitian metric by its Chern-Ricci form (2011, methodology): “Applying the higher order estimates of Gill [19], which are local, we obtain uniform C estimates for ω(t) on compact subsets of M \ E.”
The Monge–Ampère equation for non-integrable almost complex structures (2016, background): “Analogously to the result proved by Cao [7] in the Kähler case and Gill [23] in the Hermitian case, we expect that the techniques developed in this paper can be used to show that a smooth solution ϕt exists for all t > 0, and after suitable normalization converges smoothly to the solution of (1.3).”
Non-K\"ahler Calabi-Yau manifolds (2014, methodology): “Furthermore, as shown by Gill [34], on such manifolds the Chern-Ricci flow [73, 74, 77] deforms any given Hermitian metric to a Chern-Ricci flat one.”