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Tristan C. Collins

Publications and source records attributed to Tristan C. Collins.

53 records · Page 3Linked to original sources
arXiv2017

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

We prove that the zero set of a nonnegative plurisubharmonic function that solves $\det (\partial \overline{\partial} u) \geq 1$ in $\mathbb{C}^n$ and is in $W^{2, \frac{n(n-k)}{k}}$ contains no analytic sub-variety of dimension $k$ or larger. Along the way we prove an analogous result for the real Monge-Ampère equation, which is also new. These results are sharp in view of well-known examples of Pogorelov and Błocki. As an application, in the real case we extend interior regularity results to the case that $u$ lies in a critical Sobolev space (or more generally, certain Sobolev-Orlicz spaces).

math.AP↗
arXiv2016

Concavity of the Lagrangian Phase Operator and Applications

We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if $Ω$ is a compact domain in $\mathbb{R}^{n}$ or $\mathbb{C}^n$, then there exists a solution to the Dirichlet problem with right-hand side $h(x)$ satisfying $|h(x)| > (n-2)\fracπ{2}$ and boundary data $ϕ$ if and only if there exists a subsolution.

math.AP↗
arXiv2016

K stability and stability of chiral ring

We define a notion of stability for chiral ring of four dimensional N=1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N=1 field theory derived from D3 branes probing a three-fold singularity X, and show that the K stability which implies the existence of Ricci-flat conic metric on X is equivalent to the stability of chiral ring of the corresponding field theory.

hep-th↗
arXiv2015

A singular Demailly-Paun theorem

We give a numerical characterization of the Kahler cone of a possibly singular compact analytic variety which is embedded in a smooth ambient space.

math.CV↗
arXiv2015

(1,1) forms with specified Lagrangian phase: A priori estimates and algebraic obstructions

Let $(X,α)$ be a Kähler manifold of dimension n, and let $[ω] \in H^{1,1}(X,\mathbb{R})$. We study the problem of specifying the Lagrangian phase of $ω$ with respect to $α$, which is described by the nonlinear elliptic equation \[ \sum_{i=1}^{n} \arctan(λ_i)= h(x) \] where $λ_i$ are the eigenvalues of $ω$ with respect to $α$. When $h(x)$ is a topological constant, this equation corresponds to the deformed Hermitian-Yang-Mills (dHYM) equation, and is related by Mirror Symmetry to the existence of special Lagrangian submanifolds of the mirror. We introduce a notion of subsolution for this equation, and prove a priori $C^{2,β}$ estimates when $|h|>(n-2)\fracπ{2}$ and a subsolution exists. Using the method of continuity we show that the dHYM equation admits a smooth solution in the supercritical phase case, whenever a subsolution exists. Finally, we discover some stability-type cohomological obstructions to the existence of solutions to the dHYM equation and we conjecture that when these obstructions vanish the dHYM equation admits a solution. We confirm this conjecture for complex surfaces.

math.DG↗
arXiv2015

Kähler currents and null loci

We prove that the non-Kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kahler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein-Lazarsfeld-Mustata-Nakamaye-Popa. As an application, we show that finite time non-collapsing singularities of the Kahler-Ricci flow on compact Kahler 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 Kahler metrics on Calabi-Yau manifolds to the nonalgebraic case.

math.CV↗
arXiv2015

$C^{2,α}$ estimates for nonlinear elliptic equations of twisted type

We prove a priori interior $C^{2,α}$ estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex + concave. These results are particularly well suited to equations arising from elliptic regularization. As application, we obtain a new proof of an estimate of Streets-Warren on the twisted real Monge-Ampere equation.

math.AP↗
arXiv2014

Convergence of the $J$-flow on toric manifolds

We show that on a Kahler manifold whether the J-flow converges or not is independent of the chosen background metric in its Kahler class. On toric manifolds we give a numerical characterization of when the J-flow converges, verifying a conjecture of Lejmi and the second author in this case. We also strengthen existing results on more general inverse $σ_{k}$ equations on Kahler manifolds.

math.DG↗
arXiv2014

Poisson metrics on flat vector bundles over non-compact curves

Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as the dimension reduction of the Hermitian-Yang-Mills equation for holomorphic vector bundles on K3 surfaces in the large complex structure limit. We define a notion of slope stability, and show that if the flat connection D has regular singularities, and the Riemannian metric g has finite volume then E admits a Poisson metric with asymptotics determined by the parabolic structure if and only if (E,D,P) is slope polystable.

math.DG↗
arXiv2013

Remarks on the Yang-Mills flow on a compact Kahler manifold

We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X. We construct a natural barrier function along the flow, and introduce some techniques to study the blow-up of the curvature along the flow. Making some technical assumptions, we show how our techniques can be used to prove that the curvature of the evolved connection is uniformly bounded away from an analytic subvariety determined by the Harder-Narasimhan-Seshadri filtration of E. We also discuss how our assumptions are related to stability in some simple cases.

math.DG↗
arXiv2012

The twisted Kahler-Ricci flow

In this paper we study a generalization of the Kahler-Ricci flow, in which the Ricci form is twisted by a closed, non-negative (1,1)-form. We show that when a twisted Kahler-Einstein metric exists, then this twisted flow converges exponentially. This generalizes a result of Perelman on the convergence of the Kahler-Ricci flow, and it builds on work of Tian-Zhu.

math.DG↗
arXiv2012

K-Semistability for irregular Sasakian manifolds

We introduce a notion of K-semistability for Sasakian manifolds. This extends to the irregular case the orbifold K-semistability of Ross-Thomas. Our main result is that a Sasakian manifold with constant scalar curvature is necessarily K-semistable. As an application, we show how one can recover the volume minimization results of Martelli-Sparks-Yau, and the Lichnerowicz obstruction of Gauntlett-Martelli-Sparks-Yau from this point of view.

math.DG↗
arXiv2011

On the convergence of the Sasaki-Ricci flow

Given a Sasaki manifold S, we prove the Sasaki-Ricci flow converges exponentially fast to a Sasaki-Einstein metric if one exists, provided the automorphism group of the transverse holomorphic structure is trivial.

math.DG↗
arXiv2011

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 $C^{0}$ bound for the transverse scalar curvature, and a uniform $C^{1}$ bound for the transverse Ricci potential along the Sasaki-Ricci flow.

math.DG↗
arXiv2011

Stability and Convergence of the Sasaki-Ricci Flow

We introduce a holomorphic sheaf E on a Sasaki manifold and study two new notions of stability for E along the Sasaki-Ricci flow related to the `jumping up' of the number of global holomorphic sections of E at infinity. First, we show that if the Mabuchi K-energy is bounded below, the transverse Riemann tensor is bounded in C^{0} along the flow, and the C -infinity closure of the Sasaki structure under the diffeomorphism group does not contain a Sasaki structure with strictly more global holomorphic sections of E, then the Sasaki-Ricci flow converges exponentially fast to a Sasaki-Einstein metric. Secondly, we show that if the Futaki invariant vanishes, and the lowest positive eigenvalue of the d-bar Laplacian on global sections of E is bounded away from zero uniformly along the flow, then the Sasaki-Ricci flow converges exponentially fast to a Sasaki-Einstein metric.

math.DG↗
arXiv2011

Uniform Sobolev inequality along the Sasaki-Ricci flow

We prove a uniform Sobolev inequality along the Sasaki-Ricci flow. In the process, we develop the theory of basic Lebesgue and Sobolev function spaces, and prove some general results about the decomposition of the heat kernel for a class of elliptic operators on a Sasaki manifold.

math.DG↗