Tristan C. Collins

Kähler currents and null loci

We prove that the non-Kähler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kähler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein–Lazarsfeld–Mustaţă–Nakamaye–Popa. As an application, we show that finite time non-collapsing singularities of the Kähler–Ricci flow on compact Kähler manifolds always form along analytic subvarieties, thus answering a question of Feldman–Ilmanen–Knopf and Campana. We also extend the second author’s results about noncollapsing degenerations of Ricci-flat Kähler metrics on Calabi–Yau manifolds to the nonalgebraic case.

The transverse entropy functional and the Sasaki-Ricci flow

We introduce two new functionals on Sasaki manifolds, inspired by the work of Perelman, which are monotonic along the Sasaki-Ricci flow. We relate their gradient flow, via diffeomorphisms preserving the foliated structure of the manifold, to the transverse Ricci flow. Finally, when the basic first Chern class is positive, we employ these new functionals to prove a uniform

Concavity of the Lagrangian phase operator and applications

C^{0}

An extension theorem for Kähler currents with analytic singularities

bound for the transverse scalar curvature, and a uniform

C^{2,\alpha } estimates for nonlinear elliptic equations of twisted type

C^{1}

Dimension of the minimum set for the real and complex Monge–Ampère equations in critical Sobolev spaces

bound for the transverse Ricci potential along the Sasaki-Ricci flow.

A singular Demailly–Păun theorem

We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if

K-semistability for irregular Sasakian manifolds

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