arXiv research

Hermitian metrics with zero second Chern Ricci curvature are rigid and exist on specific manifolds.

Study shows uniform bounds on torsion and curvature for Chern-Ricci flow solutions.

In this note we study finite-time singularities in the Chern-Ricci flow. We show that finite-time singularities are characterized by the blow-up of the scalar curvature of the Chern connection.

Compact Kähler manifolds with positive curvature are projective and rationally connected.

Blowing up flat metrics yields balanced ones with constant curvature.

Study deforms Hermitian metrics with positive curvature.

Flow smooths Chern-Ricci-flat metrics on Hermitian manifolds.

Derivative estimates for pluriclosed flow control curvature and torsion.

Regularities and stability shown for a specific type of complex parallelizable manifolds.

Assuming local uniform bounds on the metric for a solution of the Chern-Ricci flow, we establish local Calabi and curvature estimates using the maximum principle.

Study finds criteria for surfaces with specific curvature properties.

In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Her…

Survey on metrics on non-Kähler complex manifolds.

In this note, we show that on Hopf manifold $\mathbb S^{2n-1}\times \mathbb S^1$, the non-negativity of the holomorphic bisectional curvature is not preserved along the Chern-Ricci flow.

In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of Chern-Ricci curvature will be preserved if the initial metric has non-positive bisectional curvature. As an application, we show that the canonical line bundle of a compact Hermitian manifold with nonpositive bisecti…

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative c…

The Chern sectional curvature of a Hermitian manifold is derived and related to Kähler metrics.

We construct the first and second Chern-Ricci functions on negatively curved minimal surfaces in ${\mathbb{R}}^{3}$ using Gauss curvature and angle functions, and establish that they become harmonic functions on the minimal surfaces. We prove that a minimal surface has constant first Chern-Ricci function if and only if…

In this work, we obtain existence criteria for Chern-Ricci flows on noncompact manifolds. We generalize a result by Tossati-Wienkove on Chern-Ricci flows to noncompact manifolds and at the same time generalize a result for Kahler-Ricci flows by Lott-Zhang to Chern-Ricci flows. Using the existence results, we prove that…

Study proves Kählerness criteria for Hermitian surfaces under specific curvature conditions.

Study fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.

We show that the Kähler-Ricci flow on a manifold with positive first Chern class converges to a Kähler-Einstein metric assuming positive bisectional curvature and certain stability conditions.

We determine the greatest lower bounds on the transverse Ricci curvature of compact toric Sasaki manifolds with positive basic first Chern class and with the first Chern class of the contact bundle being trivial. This is based on Wang-Zhu's and Futaki-Ono-Wang's works, and is an analogue of C. Li's work on toric Fano m…

The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.

The paper proves properties of complex surfaces and their curvature.

Bounding characteristic numbers of Riemannian manifolds via volume.

Study classifies 4D shrinkers with nonnegative Ricci curvature.

We consider dimension reduction for solutions of the Kähler-Ricci flow with nonegative bisectional curvature. When the complex dimension $n=2$, we prove an optimal dimension reduction theorem for complete translating Kähler-Ricci solitons with nonnegative bisectional curvature. We also prove a general dimension reducti…

If a normalized Kähler-Ricci flow $g(t),t\in[0,\infty),$ on a compact Kähler $n$-manifold, $n\geq 3$, of positive first Chern class satisfies $g(t)\in 2πc_{1}(M)$ and has $L^{n}$ curvature operator uniformly bounded, then the curvature operator will also uniformly bounded along the flow. Consequently the flow will conv…

The paper explores mixed curvature for Hermitian manifolds and its implications.

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli-Chern class on compact complex manifolds, and proved that the $(1,1)$ curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. …

Vanishing theorem for certain tensor fields on compact Hermitian manifolds.

A Finsler space is called Ricci-quadratic if its Ricci curvature $Ric(x,y)$ is quadratic in $y$. It is called a Berwald space if its Chern connection defines a linear connection directly on the underlying manifold $M$. In this article, we prove that a homogeneous Randers space is Ricci-quadratic if and only if it is of…

We show that Perelman's W-functional can be generalized to Sasaki-Ricci flow. When the basic first Chern class is positive, we prove a uniform bound on the scalar curvature, the diameter and a uniform $C^1$ bound for the transverse Ricci potential along the Sasaki-Ricci flow, which generalizes Perelman's results Kahler…

New metrics found on non-Kähler Calabi-Yau manifolds.

In this paper we prove first order differential Harnack estimates for positive solutions of the heat equation (in the sense of distributions) under closed Finsler-Ricci flows. We assume mild non-linearities (in terms of the Chern connection, $\mathbf{S}-$curvature and Hessian) and suitable Ricci curvature bounds throug…

Survey on Chern-Ricci flow for complex manifolds.

Uniform estimates lead to Gromov-Hausdorff limits for Hermitian minimal models.

We study the almost Kaehler geometry of adjoint orbits of non-compact real semisimple Lie groups endowed with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almost complex structure. We give explicit formulas for the Chern-Ricci form, the Hermitian scalar curvature and the Nijenhuis tensor in te…

Let $({\M}, g(t))$ be a Kähler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on $({\M}, g(t))$, and a Poincaré inequality without assuming the Ricci curvature is bound…

We continue studying a parabolic flow of almost Kähler structures introduced by Streets and Tian which naturally extends Kähler-Ricci flow onto symplectic manifolds. In the system of primarily the symplectic form, almost complex structure, Chern torsion and Chern connection, we establish new formulas for the evolutions…

The Chern-Ricci flow is an evolution equation of Hermitian metrics by their Chern-Ricci form, first introduced by Gill. Building on our previous work, we investigate this flow on complex surfaces. We establish new estimates in the case of finite time non-collapsing, anologous to some known results for the Kahler-Ricci …

In this note, we use Chern's magic form $Φ_k$ in his famous proof of the Gauss-Bonnet theorem to define a mass for asymptotically flat manifolds. It turns out that the new defined mass is equivalent to the one that we introduced recently by using the Gauss-Bonnet-Chern curvature $L_k$. Moreover, this equivalence implie…

Study of $t$-Gauduchon Ricci-flat condition under Chern-Ricci flow on non-Kähler manifolds.

The paper constructs flat metrics on orbifolds and resolutions.

The paper explores properties of Gauduchon curvature in Hermitian manifolds.

In this work, we obtain some existence results of Chern-Ricci Flows and the corresponding Potential Flows on complex manifolds with possibly incomplete initial data. We discuss the behaviour of the solution as $t\rightarrow 0$. These results can be viewed as generalization of an existence result by Giesen and Topping f…

The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.