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Jian Song

Publications and source records attributed to Jian Song.

At least 235 records · Page 13Linked to original sources
arXiv2011

The Kähler-Ricci flow on Hirzebruch surfaces

We investigate the metric behavior of the Kahler-Ricci flow on the Hirzebruch surfaces, assuming the initial metric is invariant under a maximal compact subgroup of the automorphism group. We show that, in the sense of Gromov-Hausdorff, the flow either shrinks to a point, collapses to $\mathbb{P}^1$ or contracts an exceptional divisor, confirming a conjecture of Feldman-Ilmanen-Knopf. We also show that similar behavior holds on higher-dimensional analogues of the Hirzebruch surfaces.

math.DG↗
arXiv2010

Feynman-Kac formula for heat equation driven by fractional white noise

We establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman-Kac formula is a weak solution of the stochastic heat equation. From the Feynman-Kac formula, we establish the smoothness of the density of the solution and the Hölder regularity in the space and time variables. We also derive a Feynman-Kac formula for the stochastic heat equation in the Skorokhod sense and we obtain the Wiener chaos expansion of the solution.

math.PR↗
arXiv2010

Metric flips with Calabi ansatz

We study the limiting behavior of the Kahler-Ricci flow on $\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus (m+1)})$, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to $\mathbb{P}^n$ or contracts a subvariety of codimension m+1 in Gromov-Hausdorff sense. We also show that the Kahler-Ricci flow resolves certain type of conical singularities in Gromov-Hausdorff sense.

math.DG↗
arXiv2009

Fractional martingales and characterization of the fractional Brownian motion

In this paper we introduce the notion of fractional martingale as the fractional derivative of order $α$ of a continuous local martingale, where $α\in(-{1/2},{1/2})$, and we show that it has a nonzero finite variation of order $\frac{2}{1+2α}$, under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of Lévy's characterization theorem for the fractional Brownian motion.

math.PR↗
arXiv2009

The Kahler-Ricci flow through singularities

We prove the existence and uniqueness of the weak Kahler-Ricci flow on projective varieties with log terminal singularities. It is also shown that the weak Kahler-Ricci flow can be uniquely continued through divisorial contractions and flips if they exist. We then propose an analytic version of the Minimal Model Program with Ricci flow.

math.DG↗
arXiv2009

Finite time extinction of the Kahler-Ricci flow

We investigate the limiting behavior of the unnormalized Kahler-Ricci flow on a Kahler manifold with a polarized initial Kahler metric. We prove that the Kahler-Ricci flow becomes extinct in finite time if and only if the manifold has positive first Chern class and the initial Kahler class is proportional to the first Chern class of the manifold. This proves a conjecture of Tian for the smooth solutions of the Kahler-Ricci flow.

math.DG↗
arXiv2009

Bergman metrics and geodesics in the space of Kähler metrics on toric varieties

Geodesics on the infinite dimensional symmetric space $\hcal$ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X are solutions of a homogeneous complex Monge-Ampère equation in $X \times A$, where $A \subset \C$ is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces $G_{\C}/G$. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether Monge-Ampère geodesics can be approximated by 1PS geodesics in the symmetric spaces of Bergman metrics. Phong-Sturm proved weak C^0 convergence of Bergman to Monge-Ampère geodesics on a general \kahler manifold. In this article we prove convergence in $C^2(A \times X)$ in the case of toric Kähler metrics, extending our earlier result on $\CP^1$.

math.CV↗
arXiv2009

Energy functionals and canonical Kahler metrics

Yau conjectured that a Fano manifold admits a Kahler-Einstein metric if and only if it is stable in the sense of geometric invariant theory. There has been much progress on this conjecture by Tian, Donaldson and others. The Mabuchi energy functional plays a central role in these ideas. We study the E_k functionals introduced by X.X. Chen and G. Tian which generalize the Mabuchi energy. We show that if a Fano manifold admits a Kahler-Einstein metric then the functional E_1 is bounded from below, and, modulo holomorphic vector fields, is proper. This answers affirmatively a question raised by Chen. We show in fact that E_1 is proper if and only if there exists a Kahler-Einstein metric, giving a new analytic criterion for the existence of this canonical metric, with possible implications for the study of stability. We also show that on a Fano Kahler-Einstein manifold all of the functionals E_k are bounded below on the space of metrics with nonnegative Ricci curvature.

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

Integral representation of renormalized self-intersection local times

In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the $d$% -dimensional fractional Brownian motion with Hurst parameter $H\in (0,1)$. As a consequence, we derive the existence of some exponential moments for this random variable.

math.PR↗
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↗
arXiv2008

Canonical measures and Kahler-Ricci flow

We show that the Kahler-Ricci flow on an algebraic manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Kodaira dimension. Such a canonical measure is unique and invariant under birational transformations under the assumption of the finite generation of canonical rings.

math.DG↗
arXiv2007

Test configurations, large deviations and geodesic rays on toric varieties

This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesic ray of potentials are discontinuous across certain hypersurfaces and are degenerate on certain open sets. A novelty in the analysis is the connection between Bergman metrics, Bergman kernels and the theory of large deviations.

math.DG↗
arXiv2007

Constructions of Kahler-Einstein metrics with negative scalar curvature

We show that on Kahler manifolds with negative first Chern class, the sequence of algebraic metrics introduced by H. Tsuji converges uniformly to the Kahler-Einstein metric. For algebraic surfaces of general type and orbifolds with isolated singularities, we prove a convergence result for a modified version of Tsuji's iterative construction.

math.DG↗
arXiv2007

Convergence of Bergman geodesics on CP^1

The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly approximated by geodesics in the finite dimensional spaces of Bergman metrics. We prove a stronger C^2-approximation in the special case of toric (i.e. S^1-invariant) metrics on CP^1.

math.DG↗
arXiv2006

The Moser-Trudinger inequality on Kahler-Einstein manifolds

We prove the conjecture of Tian on the strong form of the Moser-Trudinger inequality for Kahler-Einstein manifolds with positive first Chern class, when there are no holomorphic vector fields, and, more generally, when the setting is invariant under a maximal compact subgroup of the automorphism group.

math.DG↗
arXiv2006

On Donaldson's flow of surfaces in a hyperkahler four-manifold

We prove some basic properties of Donaldson's flow of surfaces in a hyperkahler 4-manifold. When the initial submanifold is symplectic with respect to one Kähler form and Lagrangian with respect to another, we show that certain kinds of singularities cannot form, and we prove a convergence result under a condition related to one considered by M.-T. Wang for the mean curvature flow.

math.DG↗