arXiv research

Study on Kähler-Ricci flow's infinite-time singularities.

The paper constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.

For the Kähler-Ricci flow on a compact Kähler manifold with semi-ample canonical line bundle, we prove the singularity type at infinity does not depend on the choice of the initial metric. We also provide new simple proofs for some existing classification results on infinite-time singularity type of the Kähler-Ricci fl…

Enhances understanding of Kähler-Ricci flow singularities.

The paper studies Yang-Mills flow on special holonomy manifolds and proves curvature bounds.

The paper constructs singularities for Lagrangian flow in Gibbons-Hawking spaces with vanishing mean curvature.

The paper proves a compactness theorem for Yang-Mills flow in higher dimensions.

We consider the general Kähler-Ricci flows which exist for all time. The zeroth order control on the flow metric potential for various infinite time singularities is the focus. The possible semi-amplness for numerically effective classes serves as the main motivation.

Consider the Kahler-Ricci flow with finite time singularities over any closed Kahler manifold. We prove the existence of the flow limit in the sense of current towards the time of singularity. This answers affirmatively a problem raised by Tian on the uniqueness of the weak limit from sequential convergence constructio…

Several results on existence and convergence of the Yang-Mills flow in dimension four are given. We show that a singularity modeled on an instanton cannot form within finite time. Given low initial self-dual energy, we then study convergence of the flow at infinite time. If an Uhlenbeck limit is anti-self-dual and has …

Ricci flow converges to Taub-NUT metric under specific conditions.

The noncompact Yamabe flow can lead to incomplete metrics over infinite time.

The paper studies singularities in a complex flow related to mean curvature.

In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divis…

We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilati…

We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so…

Study on Yang-Mills heat flow on $\mathbb{R}^4$ bundles, showing infinite time bubbling.

Infinite-time blow-up in high-dimensional mean curvature flow.

We study the collapsing behavior of the Kaehler-Ricci flow on a compact Kaehler manifold X admitting a holomorphic submersion X -> S coming from its canonical class, where S is a Kaehler manifold with dim S < dim X. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology in…

A second order linear integro-differential equation with Volterra integral operator and strong singularities at the endpoints (zero and infinity) is considered. Under limit conditions at the singular points, and some natural assumptions, the problem is a singular initial problem with limit normalizing conditions at inf…

Paper solves MFG for partially reversible investment, showing price influence and Nash equilibrium.

The Yamabe flow can blow up in infinite time with small perturbations.

Study solutions and singularities of G2-structures flows on specific manifolds.

Deep neural nets approximate random dynamical system trajectories uniformly in time.

Equivalences are known between problems of singular stochastic control (SSC) with convex performance criteria and related questions of optimal stopping, see for example Karatzas and Shreve [SIAM J. Control Optim. 22 (1984)]. The aim of this paper is to investigate how far connections of this type generalise to a non co…

Integral inequalities for holomorphic maps prove rigidity and degeneracy theorems.

We solve explicitly a two-dimensional singular control problem of finite fuel type for infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative and transient price impact. Liquidity is stochastic in that the volume effect process, which determi…

Study on flows of G2-structures on contact Calabi-Yau 7-manifolds.

Improved algorithm for optimal stopping problems reduces runtime.

Global existence and smoothing effects for reaction-diffusion equations with blowup in infinite time.

In this paper, we study the global Kähler-Ricci flow on a complete non-compact Kähler manifold. We prove the following result. Assume that $(M,g_0)$ is a complete non-compact Kähler manifold such that there is a potential function $f$ of the Ricci tensor, i.e., $R_{i\bar{j}}(g_0)=f_{i\bar{j}}.$ Assume that the quan…

Consider the problem of a government that wants to reduce the debt-to-GDP (gross domestic product) ratio of a country. The government aims at choosing a debt reduction policy which minimises the total expected cost of having debt, plus the total expected cost of interventions on the debt ratio. We model this problem as…

We establish various existence and uniqueness results for the Yang-Mills flow on cylindrical end 4-manifolds. We also show long-time existence and infinite-time convergence under certain hypotheses on the underlying data.

We consider the porous medium equation with power-type reaction terms $u^p$ on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If $p>m$, small data give rise to global-in-time solutions while solutions associated to large data blow up in finite t…

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the …

Study mean curvature flow of high codimension submanifolds in complex projective space.

We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Lévy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting,…

We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics, on Inoue surfaces. These are non-Kahler compact complex surfaces of type Class VII. We show that, after an initial conformal change, the flow always collapses the Inoue surface to a circle at infinite time, in the sense of Gromov-Hausdorff…

These notes are the first half of the contents of the course given by the second author at the Bachelier Seminar (February 8-15-22 2008) at IHP. They also correspond to topics studied by the first author for her Ph.D.thesis.

Project infinite time series graphs to finite marginal models using number theory.

Study optimal liquidation with incomplete trend information and multiplicative price impact.

New tools extend weak approximation of SGD algorithms to infinite time horizon.

Consider the problem of a central bank that wants to manage the exchange rate between its domestic currency and a foreign one. The central bank can purchase and sell the foreign currency, and each intervention on the exchange market leads to a proportional cost whose instantaneous marginal value depends on the current …

We show an energy convexity along any harmonic map heat flow with small initial energy and fixed boundary data on the unit 2-disk. In particular, this gives an affirmative answer to a question raised by W. Minicozzi asking whether such harmonic map heat flow converges uniformly in time strongly in the W^{1,2}-topology,…

Paper shows Ricci flow with surgery converging to Taub-NUT metric.

We study solvency of insurers in a comprehensive model where various economic factors affect the capital developments of the companies. The main interest is in the impact of real growth to ruin probabilities. The volume of the business is allowed to increase or decrease. In the latter case, the study is focused on run-…

For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwi…

Curves converge to circles under length constraints.