arXiv research

The paper studies the modified J-equation on Kähler manifolds.

The paper proves conditions for solutions of $J$-equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions.

Characterizes solvability of J-equation on Kähler surfaces with singularities.

Investigates $J$-equation on holomorphic vector bundles over Kähler manifolds.

New criterion for solving inverse Hessian equations, including J-equation.

Proof that certain complex equations have only simple solutions.

This paper proves a Nakai-Moishezon criterion for complex Hessian equations.

In this paper, we prove that for any Kähler metrics $ω_0$ and $χ$ on $M$, there exists $ω_\varphi=ω_0+\sqrt{-1}\partial\bar\partial\varphi>0$ satisfying the J-equation $\mathrm{tr}_{ω_\varphi}χ=c$ if and only if $(M,[ω_0],[χ])$ is uniformly J-stable. As a corollary, we can find many constant scalar curvature Kähler met…

Solves open problems for fully nonlinear elliptic equations on manifolds.

Introduces a new PDE involving differential forms for Kähler geometry.

Characterizes solutions to Z-critical equations on surfaces using effective conditions.

Proves solvability of general inverse σ_k equations with constant coefficients.

Proves smooth solutions for generalised Monge-Ampère equations on projective manifolds.

In this paper we provide an explicit formula for the optimal lower bound of Donaldson's J-functional, in the sense of finding explicitly the optimal constant in the definition of coercivity, which always exists and takes negative values in general. This constant is positive precisely if the J-equation admits a solution…

Solves complex Hessian equations in unstable cases, proving unique canonical solutions with singularities.

Study of J-flow on Kähler manifolds confirms energy properness.

Study proves estimate for Hessian quotient equations on 2D Riemannian manifolds.

In this paper, we study the deformed Hermitian-Yang-Mills equation on compact Kähler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with respect to the background metric if there exists a positive constant $C$ such that $-\…

Paper confirms conjecture for projective manifolds in supercritical phase.

Study of singularity formation in dHYM flow on a blown-up CP^3.

The paper characterizes when numerical criteria for PDE solvability fail and provides effective criteria for existence.

Proves convexity of level sets of general inverse σ_k equations.

Invariants of 3-manifolds using modified Hopf G-coalgebra.

Develops methods to solve complex and real Hessian equations.

We study the properties of Modified Riemann extensions evolving under Ricci flow. We obtain the necessary and sufficient condition for modified Riemann extension under Ricci flow to stay as modified Riemann extension. We also discuss the properties of the curvature tensors under Ricci flow.

Sharp eigenvalue bounds and splitting for modified Ricci flow.

Modified Hennings invariant defined using quantum groups and integrals.

Modified ReLU networks improve regression estimation rates.

The modified J-flow with Calabi ansatz shows convergence or blow-up behavior based on topological constants.

We modify the Laplacian coflow of co-closed G2-structures - $\frac{d}{dt}ψ=Δψ$ where $ψ$ is the closed dual 4-form of a $G_{2}$-structure $\varphi$. The modified flow is now parabolic in the direction of closed forms upto diffeomorphisms. We then prove short time existence and uniqueness of solutions to the modified f…

Study new Einstein-like metrics and their properties.

The paper proves that certain modified conformal vector fields are trivial on compact and non-compact manifolds.

Modified curve shortening flow constructs $λ$-Angenent curve.

We introduce the conical Kähler-Ricci flow modified by a holomorphic vector field. We construct a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.

Study lower bounds on modified K-energy on Fano manifolds with Kähler-Ricci solitons.

We construct finite-gap solutions to the modified Novikov-Veselov equations, describe their spectral properties and the reduction to the modified Korteweg--de Vries equation and explain its relation to soliton deformations of tori and the Willmore conjecture.

We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the $τ$ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete …

New method reveals insights about stochastic optimization methods using modified equations.

Formula derived for torsion of modified Dirac operator.

In this paper, we give two Lichnerowicz type formulas for modified Novikov operators. We prove KastlerKalau-Walze type theorems for modified Novikov operators on compact manifolds with (resp.without) boundary. We also compute the spectral action for Witten deformation on 4-dimensional compact manifolds.

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.

We give a characterization of relative Ding stable toric Fano manifolds in terms of the behavior of the modified Ding functional. We call the corresponding behavior of the modified Ding functional the pseudo-boundedness from below. We also discuss the pseudo-boundedness of the Ding / Mabuchi functional of general Fano …

The paper proves two theorems for modified Novikov operators under conformal perturbations.

We give estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford algebra element associated to a multi-degree form. In particular such estimates are presented for modified Dirac operators with a $k$-degree form $0\leq k\leq…

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and give necessary and sufficient conditions so that a modified Riemannian extension is Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3,3) whose Jacobi operators have non-trivial Jordan normal …

Study on manifolds with density using modified Hessians for curvature comparison.

In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified $K$-energy is proper f…

New dynamics for SGD in small learning rate regime.