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D. H. Phong

Publications and source records attributed to D. H. Phong.

At least 19 recordsLinked to original sources
arXiv2015

Convergence of the conical Ricci flow on S2 to a soliton

In our previous work [PSSW], we showed that the Ricci flow on S^2 whose initial metric has conical singularities \sum_{j=1}^k β_j[p_j] converges to a constant curvature metric with conic singularities (in the stable and semi-stable cases) or to a gradient shrinking soliton with conical singularities (in the unstable case). The purpose of this note is to show that in the unstable case, that is, the case where β_k>β_k'=\s_{j<k}β_j, that the limiting metric is the unique shrinking soliton with cone singularity β_k[p_\infty]+β_k'[q_\infty]. This verifies the prediction made in [PSSW].

math.DG↗
arXiv2014

The Ricci flow on the sphere with marked points

The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is given. The semi-stable and unstable cases are new, and it is shown that the flow converges in the Gromov-Hausdorff topology to a limiting metric space which is also a 2-sphere, but with different marked points and hence a different complex structure. The limiting metric space carries a unique conical constant curvature metric in the semi-stable case, and a unique conical shrinking gradient Ricci soliton in the unstable case.

math.DG↗
arXiv2012

Degeneration of Kahler-Ricci solitons on Fano manifolds

We consider the space KR(n,F) of Kahler-Ricci solitons on n-dimensional Fano manifolds with Futaki invariant bounded by F. We prove a partial C^0 estimate for KR(n,F) as a generalization of the recent work of Donaldson-Sun for Fano Kahler-Einstein manifolds. In particular, any sequence in KR(n,F) has a convergent subsequence in the Gromov-Hausdorff topology to a Kahler- Ricci soliton on a Q-Fano variety with log terminal singularities.

math.DG↗
arXiv2012

Complex Monge Ampere Equations

This is a survey of some of the recent developments in the theory of complex Monge-Ampere equations. The topics discussed include refinements and simplifications of classical a priori estimates, methods from pluripotential theory, variational methods for big cohomology classes, semiclassical constructions of solutions of homogeneous equations, and envelopes.

math.DG↗
arXiv2012

On the singularities of the pluricomplex Green's function

It is shown that, on a compact Kahler manifold with boundary, the singularities of the pluricomplex Green's function with multiple poles can be prescribed to be of the form $\log\sum_{j=1}^n|f_j(z)|^2$ at each pole, where $f_j(z)$ are arbitrary local holomorphic functions with the pole as their only common zero. The proof is a combination of blow-ups and recent a priori estimates for the degenerate complex Monge-Ampere equation, and particularly the $C^1$ estimates away from a divisor.

math.DG↗
arXiv2010

Partial Legendre transforms of non-linear equations

The partial Legendre transform of a non-linear elliptic differential equation is shown to be another non-linear elliptic differential equation. In particular, the partial Legendre transform of the Monge-Ampère equation is another equation of Monge-Ampère type. In 1+1 dimensions, this can be applied to obtain uniform estimates to all orders for the degenerate Monge-Ampère equation with boundary data satisfying a strict convexity condition.

math.AP↗
arXiv2010

A maximum rank problem for degenerate elliptic fully nonlinear equations

The solutions to the Dirichlet problem for two degenerate elliptic fully nonlinear equations in $n+1$ dimensions, namely the real Monge-Ampère equation and the Donaldson equation, are shown to have maximum rank in the space variables when $n \leq 2$. A constant rank property is also established for the Donaldson equation when $n=3$.

math.AP↗
arXiv2009

Regularity of geodesic rays and Monge-Ampere equations

It is shown that the geodesic rays constructed as limits of Bergman geodesics from a test configuration are always of class $C^{1,α}, 0<α<1$. An essential step is to establish that the rays can be extended as solutions of a Dirichlet problem for a Monge-Ampere equation on a Kaehler manifold which is compact.

math.DG↗
arXiv2009

The Dirichlet problem for degenerate complex Monge-Ampere equations

The Dirichlet problem for a Monge-Ampere equation corresponding to a nonnegative, possible degenerate cohomology class on a Kaehler manifold with boundary is studied. C^{1,α} estimates away from a divisor are obtained, by combining techniques of Blocki, Tsuji, Yau, and pluripotential theory. In particular, C^{1,α} geodesic rays in the space of Kaehler potentials are constructed for each test configuration

math.DG↗
arXiv2008

The modified Kähler-Ricci flow and solitons

We investigate the Kähler-Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a Kähler-Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy.

math.DG↗
arXiv2008

Lectures on Stability and Constant Scalar Curvature

An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite-dimensional GIT, and conditions on the closure of orbits of almost-complex structures under the diffeomorphism group. Related analytic methods are also discussed, including estimates for energy functionals, Tian-Yau-Zelditch approximations, estimates for moment maps, complex Monge-Ampere equations and pluripotential theory, and the Kaehler-Ricci flow

math.DG↗
arXiv2008

The Kähler-Ricci flow and the $\bar\partial$ operator on vector fields

The limiting behavior of the normalized Kähler-Ricci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi K-energy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabuchi K-energy is bounded from below and if the lowest positive eigenvalue of the $\bar\partial^\dagger \bar\partial$ operator on smooth vector fields is bounded away from 0 along the flow, then the metrics converge exponentially fast in $C^\infty$ to a Kähler-Einstein metric.

math.DG↗
arXiv2007

Two-Loop Superstrings VII, Cohomology of Chiral Amplitudes

The relation between superholomorphicity and holomorphicity of chiral superstring N-point amplitudes for NS bosons on a genus 2 Riemann surface is shown to be encoded in a hybrid cohomology theory, incorporating elements of both de Rham and Dolbeault cohomologies. A constructive algorithm is provided which shows that, for arbitrary N and for each fixed even spin structure, the hybrid cohomology classes of the chiral amplitudes of the N-point function on a surface of genus 2 always admit a holomorphic representative. Three key ingredients in the derivation are a classification of all kinematic invariants for the N-point function, a new type of 3-point Green's function, and a recursive construction by monodromies of certain sections of vector bundles over the moduli space of Riemann surfaces, holomorphic in all but exactly one or two insertion points.

hep-th↗
arXiv2007

On the regularity of geodesic rays associated to test configurations

Geodesic rays of class C^{1,1} are constructed for any test configuration of a positive line bundle L on X using resolution of singularities. The construction reduces to finding a subsolution of the corresponding Monge-Ampere equation. Geometrically, this is accomplished by the use a positive line bundle on the resolution which is trivial outside of the exceptional divisor.

math.DG↗
arXiv2007

Deligne pairings and the Knudsen-Mumford expansion

Let $X\to B$ be a proper flat morphism between smooth quasi-projective varieties of relative dimension $n$, and $L\to X$ a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for $\det (π_* L^k)$ in terms of Deligne pairings of $L$ and the relative canonical bundle $K$. This generalizes a theorem of Deligne which holds for families of relative dimension one. As a corollary, we show that when $X$ is smooth (or, more generally, if $X$ fits in a smooth family), the line bundle associated to $X\to B$, which was introduced by the first and third authors, coincides with the CM bundle defined by Paul-Tian. In a second corollary, we establish asymptotics for the K-energy along Bergman rays, generalizing a formula obtained by Paul-Tian.

math.DG↗