Prove long-time existence of pluriclosed flow on certain fibrations
problem Long-time existence of pluriclosed flow on fibrations
method General theorem on holomorphic submersions
result Long-time existence of pluriclosed flow on nilmanifolds, almost-abelian solvmanifolds, and certain complex surfaces
Researchers prove long-time existence for two landmark Brownian motion.
problem Proving long-time existence of Brownian motion on configurations of two landmarks.
method Classification and analysis of long-time existence for configurations of exactly two landmarks, using a radial kernel.
result For configurations of exactly two landmarks, long-time existence is possible for certain kernels, but not for others.
Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.
problem Existence and convergence of twisted Calabi flow on compact Kähler manifolds.
method Analysis of a family of twisted Calabi flows connecting J-flow and Calabi flow, showing long-time existence and convergence to cscK metrics.
result Long-time existence and convergence of twisted Calabi flow to cscK metrics, implying openness of continuity method.
New criteria for long-time existence of parabolic flow from 11D supergravity.
problem Establishing long-time existence of a parabolic flow from 11D supergravity.
method Using only Ricci curvatures and their derivatives, along with 4-forms, to establish long-time existence.
result A new criteria for long-time existence of the parabolic flow.
The anomaly flow on a complex 3-fold is studied with integral Shi-type estimates and long-time existence conditions.
problem Long-time existence of the anomaly flow on a compact complex 3-fold.
method Integral Shi-type estimates adapted from integration-by-parts arguments, with a smallness condition on the slope parameter.
result Long-time existence of the anomaly flow on a compact complex 3-fold under a smallness condition on the slope parameter.
Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.
problem Proving long-time existence and topological rigidity for manifolds with pinched scale-invariant integral curvature.
method Proves long-time existence of Ricci flow for manifolds with bounded curvature and pinched scale-invariant integral curvature, converging to a flat metric.
result Flow converges to a flat metric, implying topological rigidity of the manifold.
Develops a new parabolic equation for surfaces, proving long-time existence and convergence.
problem Extending elliptic equations to parabolic settings for surfaces.
method Introduces a parabolic analogue of the elliptic split-type Monge-Ampère equation.
result Proves long-time existence and convergence conditions for the new equation.
Existence of balanced embedding proved for complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifolds into infinite-dimensional projective space.
method Gradient flow in a Hilbert space, long-time existence established by perturbation, convergence depends on a priori bounds.
result Existence of balanced embedding proved in a model case.
Introduces generalized Yamabe flows with long-time existence and convergence results.
problem Yamabe flow and its limitations.
method Introduces a family of conformal flows generalizing the classical Yamabe flow and proves long-time existence and convergence.
result Long-time existence and convergence for a large class of generalized Yamabe flows.
Study shows long-term flow on special manifolds with positive Yamabe constant.
problem Analyzing long-time behavior of Yamabe flow on singular spaces.
method Formulated axioms for long-time existence, used parabolic Moser iteration for bounds.
result Established long-time existence of normalized Yamabe flow on specified manifolds.
The (α,β)-Ricci-Yamabe flow exists on closed manifolds.
problem Existence of solutions to the (α,β)-Ricci-Yamabe flow. method Showed short time existence and established long time existence theorems.
result Existence of smooth solutions to the (α,β)-Ricci-Yamabe flow on closed manifolds. Proves smooth solution uniqueness and long-term existence for a parabolic equation on a complex manifold.
problem Existence and uniqueness of solutions to a parabolic equation on compact complex manifolds.
method Uses parabolic Donaldson's equation to prove existence and uniqueness of smooth solutions.
result Smooth solutions to the parabolic Donaldson's equation on compact complex manifolds exist and are unique for all time.
In this paper we study Inverse Mean Curvature Flow (IMCF) on manifolds that are conformal to a warped product manifold. To this end, we show how the gradient conformal vector field in warped product manifolds is related to the conformal vector field on the conformal metric and use this to gain control of the flow in or…
Given a closed 3-manifold with an initial Riemannian metric of negative sec- tional curvature, we consider the cross curvature flow an evolution equation of metric on M3. We prove long-time existence of a solution to the cross curvature flow via the maximum principle theorem. Besides, we demonstrate the solution exists…
Study shows Ricci flow's convergence and harmonic map heat flow's long-time existence.
problem Analyzing convergence of Ricci flow and harmonic map heat flow.
method Established long-time existence of harmonic map heat flow between Ricci flow and shrinker.
result Ricci flow converges exponentially to compact integrable shrinkers and at singularities modelled on the shrinker.
We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with larg…
We recast the Calabi flow in DeGiorgi's language of minimizing movements. We establish the long time existence of minimizing movements for K-energy with arbitrary initial condition. Furthermore we establish some a priori regularity of these solutions, and that sufficiently regular minimizing movements are smooth soluti…
Study long-term asset liquidation behavior with external flows.
problem Investigate optimal liquidation in presence of external flows.
method Convergence analysis of BSDEs for value function and strategy.
result Long-term liquidation may not occur due to external flows.
Paper proves curvature estimates for a specific flow on Kähler manifolds.
problem Proving local curvature estimates for a specific flow on Kähler manifolds.
method Proves local curvature estimates for the κ-LYZ flow over Kähler manifolds. result Generalizes the long time existence of the flow.
Study shows long-term solutions for complex equations on curved spaces.
problem Long-term behavior of solutions to fully non-linear parabolic equations on Hermitian manifolds.
method Used general assumptions and derived a Harnack inequality for the linearized equation.
result Proved the long-time existence and convergence of solutions.
We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space Rn,m, which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of…
Study solves a generalized Christoffel-Minkowski problem using curvature flow.
problem Generalization of the Lp-Christoffel-Minkowski problem. method Anisotropic curvature flow to derive long-time existence and smooth solutions.
result Existence of smooth solutions for c=1 under certain initial data. Study an anisotropic capillary flow to solve capillary Orlicz-Minkowski problem.
problem Solve capillary Orlicz-Minkowski problem without evenness assumption.
method Analyze an anisotropic capillary Gauss curvature flow to prove convergence and establish existence.
result Establish existence result for capillary Orlicz-Minkowski problem without evenness assumption.
Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.
problem Evolution of curves with fixed length and clamped boundary conditions.
method Negative L2-gradient flow of elastic energy, existence, parabolic smoothing, constrained Lojasiewicz-Simon gradient inequality. result Convergence to a critical point as time tends to infinity.
In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a recently result of…
We show that K-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kahler class long time solutions of Calabi flow minimize both K-energy and Calabi energy. Lastly, by applying convergence results from the theory of minimizing …
Extends Kyle model to multiple traders with different time-preference coefficients.
problem Existence and convergence of discrete-time Kyle models with multiple insiders.
method Extends Basak and Cuoco's model to include traders with different time-preference coefficients.
result Parameter restrictions ensure the existence of a Radner equilibrium and long-term survival of traders.
The paper studies how certain surfaces evolve over time.
problem Evolution of specific types of surfaces in high dimensions.
method Approximation by simpler problems to prove long-term existence.
result Long-time existence of the Hα-flow for specified surfaces. Extended model ensures long-term survival of traders in limited stock market participation.
problem Limited stock market participation and survival of traders over long periods.
method Extended Basak and Cuoco (1998) model with different time-preference coefficients.
result Parameter restrictions ensure long-term survival of traders.
Study of prescribed mean curvature flow on noncompact hypersurfaces in Lorentz manifolds.
problem Short time existence and long time existence of prescribed mean curvature flow on noncompact spacelike hypersurfaces.
method Finding sufficient conditions for short time existence and discussing long time existence and convergence.
result Sufficient conditions for short time existence of prescribed mean curvature flow on noncompact spacelike hypersurfaces.
Gradient flows for knot energies ensure long-term existence of knotted loops.
problem Ensuring long-term existence of knotted loops under various energies.
method Banach gradient flows, curves of maximal slope, logarithmic strain control.
result Established long-time existence of gradient flows for knot energies.
Study proves long-term solutions to a specific equation on hyperKähler manifolds.
problem Proving long-term existence and uniqueness of solutions to a parabolic quaternionic Monge-Ampère equation.
method Proved long-term existence and uniqueness using parabolic quaternionic Monge-Ampère type equation.
result Solution converges smoothly to the unique solution of the Monge-Ampère equation.
Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.
problem Proving Lipschitz regularity for harmonic map heat flows into CAT(0) spaces.
method Elliptic approximation method
result Every weak solution of the harmonic map heat flow into CAT(0) spaces is Lipschitz continuous in both space and time.
We define functionals generalising the Seiberg-Witten functional on closed spinc manifolds, involving higher order derivatives of the curvature form and spinor field. We then consider their associated gradient flows and, using a gauge fixing technique, are able to prove short time existence for the flows. We then pr…
The paper studies a flow of convex hypersurfaces using anisotropic curvature functions.
problem Deforming convex hypersurfaces in Euclidean space.
method Fully nonlinear curvature flow involving k-th elementary symmetric function and support function.
result Long-time existence and convergence of the flow under certain assumptions.
We consider the (n-1)-plurisubharmonic flow, suggested by Tosatti-Weinkove, and prove a formula for its maximal time of existence. This includes estimates that will be useful in further investigating the flow.
Long short-term memory network outperforms seasonal model in JSE Top 40 forecasting.
problem Comparing neural network performance to traditional models in financial forecasting.
method Used long short-term memory network for JSE Top 40 return data forecasting.
result Long short-term memory network outperforms seasonal model in forecasting.
TimeBridge addresses non-stationarity in long-term time series forecasting.
problem Non-stationarity in multivariate time series leads to spurious regressions and obscures long-term relationships.
method TimeBridge segments series into patches, applying Integrated Attention for short-term non-stationarity and Cointegrated Attention for long-term cointegration.
result TimeBridge achieves state-of-the-art performance in both short-term and long-term forecasting.
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.
The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths.
method Introducing combinatorial Calabi flows and proving their long time existence and global convergence.
result Proves the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary.
Improved GRU model with weighted time-delay feedback for long-term dependencies.
problem Modeling long-term dependencies in sequential data.
method Introducing a gated recurrent unit (GRU) with a weighted time-delay feedback mechanism.
result τ-GRU outperforms state-of-the-art models on various tasks.
Study shows uniform bounds on torsion and curvature for Chern-Ricci flow solutions.
problem Understanding singularity types in long-time solutions of Chern-Ricci flow.
method Extended results from Kähler-Ricci flow to Chern-Ricci flow, focusing on uniform bounds on torsion and curvature.
result Uniform bounds on torsion and curvature for solutions starting from metrics of the same ∂∂ˉ class. We investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in Hn+1 and show long time existence of the flow. Along the way many important local estimates as well as global estimates are obtained. In addition,…
Study curve shortening flows on specific surfaces, proving properties and existence.
problem Analyzing curve shortening flows on rotational surfaces with negative Gauss curvatures.
method Assume negative Gauss curvatures and conditions on Gauss curvature and curve curvature. Prove curve remains a graph and establish flow properties.
result Prove the curve remains a graph over parallels and establish long-time existence of the flow.
This paper proposes a framework to predict long-term trends and short-term fluctuations in multivariate time series.
problem Existing prediction methods often ignore the distinction between long-term trends and short-term fluctuations.
method The paper introduces a MTS forecasting framework that uses both original time series and its first difference to capture long-term trends and short-term fluctuations.
result The proposed method improves forecasting performance by using more supervision information.
This article is a continuation of earlier work [R.L. Huang and Y.H. Ye, On the second boundary value problem for a class of fully nonlinear flows I, to appear in International Mathematics Research Notices], where the long time existence and convergence were given on some general parabolic type special Lagrangian equati…
Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time existence.
In this paper, we study the evolution of submannifold moving by mean curvature minus a external force field. We prove that the flow has a long-time smooth solution for all time under almost optimal conditions. Those conditions are that the second fundamental form on the initial submanifolds is not too large, the extern…