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In this paper, we prove an optimal systolic inequality and the corresponding rigidity in the equality case on closed manifolds with positive bi-Ricci curvature, which generalizes the work of Bray-Brendle-Neves. The proof is given in all dimensions based on the method of minimal surfaces under the Generic Regularity Hypothesis, which is known to be true up to dimension ten., Comment: 23 pages, no figure

In this work, we investigate compact K\"ahler manifolds with non-negative or quasi-positive mixed curvature coming from a linear combination of the Ricci and holomorphic sectional curvature, which covers various notions of curvature considered in the literature. Specifically, we prove a splitting theorem, analogous to the Cheeger-Gromoll splitting theorem, for complete K\"ahler manifolds with non-negative mixed curvature containing a line, and then establish a structure theorem for compact K\"ahler manifolds with non-negative mixed curvature. We also show that the Hodge numbers of compact K\"ahler manifolds with quasi-positive mixed curvature must vanish. Both results are based on the conformal perturbation method., Comment: 29 pages

In this work, we study the short-time existence theory of Ricci-DeTurck flow starting from rough metrics which satisfy a Morrey-type integrability condition. Using the rough existence theory, we show the preservation and improvement of distributional scalar curvature lower bounds provided the singular set for such metrics is not too large. As an application, we use the Ricci flow smoothing to study the removable singularity for scalar curvature rigidity in the compact case under Morrey regularity conditions. Our result supplements those of Jiang-Sheng-Zhang., Comment: 23 pages; abstract updated

The classical Llarull theorem states that a smooth metric on $n$-sphere cannot have scalar curvature no less than $n(n-1)$ and dominate the standard spherical metric at the same time unless it is the standard spherical metric. In this work, we prove that Llarull's rigidity theorem holds for $L^{\infty}$ metrics on spheres with finitely many points punctured. This is related to a question of Gromov., Comment: printing mistakes corrected, 10 pages

We consider the K\"ahler-Ricci flow on compact K\"ahler manifolds with semiample canonical bundle and intermediate Kodaira dimension, and show that the flow collapses to a canonical metric on the base of the Iitaka fibration in the locally smooth topology and with bounded Ricci curvature away from the singular fibers. This follows from an asymptotic expansion for the evolving metrics, in the spirit of recent work of the first and third-named authors on collapsing Calabi-Yau metrics, and proves two conjectures of Song and Tian., Comment: 89 pages

Motivated by recent work of Deruelle-Schulze-Simon, we study complete weakly PIC1 Ricci flows with Euclidean volume growth coming out of metric cones. We show that such a Ricci flow must be an expanding gradient Ricci soliton, and as a consequence, any metric cone at infinity of a complete weakly PIC1 K\"ahler manifold with Euclidean volume growth is biholomorphic to complex Euclidean space in a canonical way., Comment: 21 pages, minor changes, all comments are welcome

In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between certain positively curved manifolds is either homotopy trivial, Riemannian submersion, local isometry or isometric immersion. This implies that an area non-increasing self map of $\mathbb{CP}^n$, $n\ge 2$ is either homotopically trivial or is an isometry. This confirms a speculation of Tsai-Tsui-Wang. We also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci flow to establish related results on manifolds with positive $1$-isotropic curvature., Comment: 35 pages, minor changes in abstract and introduction, all comments are welcome

In this work, we show that complete non-compact manifolds with non-negative Ricci curvature, Euclidean volume growth and sufficiently small curvature concentration are necessarily flat Euclidean space., Comment: 10 pages

Motivated by the recent work of Tsai-Tsui-Wang, we consider the rigidity of map between compact manifolds with positive curvature. We show that distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial, which partially solves a question of Tsai-Tsui-Wang about weakly contracting maps. The rigidity result also holds on a wider class of manifolds with positive curvature and weaker contracting property on the map in between distance non-increasing and area non-increasing. This is based on the harmonic map heat flow., Comment: result improved, minor revision, 13 pages, all comments are welcome

In this work, we use the Ricci flow approach to study the gap phenomenon of Riemannian manifolds with non-negative curvature and sub-critical scaling invariant curvature decay. The first main result is a quantitative Ricci flow existence theory without non-collapsing assumption. We use it to show that complete non-compact manifolds with non-negative complex sectional curvature and sufficiently small average curvature decay are necessarily flat. The second main result concerns three-manifolds with non-negative Ricci curvature of quadratic decay. By combining our newly established curvature estimate and method in K\"ahler geometry, we show that if the curvature decays slightly faster even in average sense, the manifold must be flat. This strengthens a result of Reiris. In the compact case, we use the Ricci flow regularization to generalize the celebrated Gromov-Ruh Theorem in this direction., Comment: 35 pages, minor changes

In this work, we extend the existence theory of non-collapsed Ricci flows from point-wise curvature lower bound to Kato-type lower bound. As an application, we prove that compact three dimensional non-collapsed strong Kato limit space is homeomorphic to a smooth manifold. The result also holds in higher dimension under uniform strong Kato lower bound of 1-isotropic curvature. We also use the Ricci flow smoothing to study stability problem in scalar curvature geometry., Comment: printing mistakes corrected, proof expanded, symbols unified, 21 pages

Recently it has been proved (Lee-Topping 2022, Deruelle-Schulze-Simon 2022, Lott 2019) that three-dimensional complete manifolds with non-negatively pinched Ricci curvature must be flat or compact, thus confirming a conjecture of Hamilton. In this paper we generalise our work on the existence of Ricci flows from non-compact pinched three-manifolds in order to prove a higher-dimensional analogue. We construct a solution to Ricci flow, for all time, starting with an arbitrary complete non-compact manifold that is PIC1 pinched. As an application we prove that any complete manifold of non-negative complex sectional curvature that is PIC1 pinched must be flat or compact., Comment: Some supporting results generalised to make them more applicable in other papers; extra references added; some additional explanation added; some notation simplified; typos corrected. Results and proofs remain the same, although some theorem numbers have changed

Motivated by the recent progress on positive mass theorem for asymptotically flat manifolds with arbitrary ends and the Gromov's definition of scalar curvature lower bound for continuous metrics, we start a program on the positive mass theorem for asymptotically flat manifolds with $C^0$ arbitrary ends. In this work as the first step, we establish the positive mass theorem of asymptotically flat manifolds with $C^0$ arbitrary ends when the metric is $W^{1,p}_{\mathrm{loc}}$ for some $p\in(n,\infty]$ and is smooth away from a non-compact closed subset with Hausdorff dimension $n-\frac{p}{p-1}$. New techniques are developed to deal with non-compactness of the singular set., Comment: 30 pages, 1 figure

In a recent work of Brendle-Hirsch-Johne, a notion of intermediate curvature was introduced to extend the classical non-existence theorem of positive scalar curvature on torus to product manifolds. In this work, we study the rigidity when the intermediate curvature is only non-negative when the ambient dimension is at most $5$., Comment: 18 pages

Motivated by the Lipschitz rigidity problem in scalar curvature geometry, we prove that if a closed smooth spin manifold admits a distance decreasing continuous map of non-zero degree to a sphere, then either the scalar curvature is strictly less than the sphere somewhere or the map is a distance isometry. Moreover, the property also holds for continuous metrics with scalar curvature lower bound in some weak sense. This extends a result in the recent work of Cecchini-Hanke-Schick and answers a question of Gromov. The method is based on studying the harmonic map heat flow coupled with the Ricci flow from rough initial data to reduce the case to smooth metrics and smooth maps so that results by Llarull can be applied., Comment: 17 pages

In this work, we construct distance like functions with integral hessian bound on manifolds with small curvature concentration and use it to construct Ricci flows on manifolds with possibly unbounded curvature. As an application, we study the geometric structure of those manifolds without bounded curvature assumption. In particular, we show that manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors $n$-regular and small curvature concentration are topologically Euclidean., Comment: 31 pages

In this paper, we study the hypercritical deformed Hermitian-Yang-Mills equation on compact K\"ahler manifolds and resolve two conjectures of Collins-Yau.

Motivated by the recent work of Lamm and Simon, in this work we study the short-time existence theory of Ricci-Deturck flow starting from rough metrics which are bi-Lipschitz and have small local scaling invariant gradient concentration. As applications, we use it to study several stability problems related to the scalar curvature., Comment: 36 pages, printing mistakes corrected, added result on positive mass Theorem on singular metrics

We show that every complete non-compact three-manifold with non-negatively pinched Ricci curvature admits a complete Ricci flow solution for all positive time, with scale-invariant curvature decay and preservation of pinching. Combining with recent work of Lott and Deruelle-Schulze-Simon gives a proof of Hamilton's pinching conjecture without additional hypotheses., Comment: v2 adds a section outlining the strategy of the proof. A few additions have been made to the introduction. Theorems and proofs remain the same

We show that any Riemannian metric conformal to the round metric on $S^n$, for $n\geq 4$, arises as a limit of a sequence of Riemannian metrics of positive scalar curvature on $S^n$ in the sense of uniform convergence of Riemannian distance. In particular, non-negativity of scalar curvature is not preserved under such limits., Comment: Version 2 adds some references and a brief discussion of the 3D situation

On a complete $p$-nonparabolic $3$-dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper $p$-Green function along its level sets for $1

We consider the problem of when a smooth Ricci flow, for positive time, that attains smooth initial data in a weak sense must be smooth down to the initial time. We obtain curvature estimates for an example where this fails. We prove a positive result in the case that the flow satisfies a lower IC1 curvature bound, equivalent to a lower Ricci bound in three dimensions. As an application, we prove that Gromov-Hausdorff limits of WPIC1 manifolds are WPIC1., Comment: Accepted to IMRN

Motivated by the torus stability problem, in this work we study K\"ahler metrics with almost non-negative scalar curvature on complex torus. We prove that after passing to a subsequence, non-collapsing sequence of K\"ahler metrics with almost non-negative scalar curvature will converge to flat torus weakly., Comment: 13 pages, minor mistakes corrected

A classical theorem in conformal geometry states that on a manifold with non-positive Yamabe invariant, a smooth metric achieving the invariant must be Einstein. In this work, we extend it to the singular case and show that in all dimension, if a continuous metric is smooth outside a compact set of high co-dimension and achieves the Yamabe invariant, then the metric is Einstein away from the singularity and can be extended to be smooth on the manifold in a suitable sense. As an application of the method, we prove a Positive Mass Theorem for asymptotically flat manifolds with analogous singularities., Comment: minor revision, printing mistakes corrected

In this work, we consider sequences of $C^2$ metrics which converge to a $C^2$ metric in $C^0$ sense. We show that if the scalar curvature of the sequence is almost non-negative in the integral sense, then the limiting metric has scalar curvature lower bound in point-wise sense., Comment: final version, 11 pages, minor mistakes corrected

In this work, we study the deformed Hermitian-Yang-Mills equation on compact K\"ahler manifold. We introduce the notions of coerciveness and properness of the $\mathcal{J}$-functional on the space of almost calibrated $(1,1)$-forms and show that they are both equivalent to the existence of solutions to the hypercritical deformed Hermitian-Yang-Mills equation., Comment: 45 pages

Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^\infty)$ on a compact manifold $M^n$ ($n\ge 3$) with negative Yamabe invariant $\sigma(M)$. It is well-known that if $g$ is a smooth metric on $M$ with unit volume and with scalar curvature $R(g)\ge \sigma(M)$, then $g$ is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles $\leq 2\pi$ along codimension-2 submanifolds. We also show in three dimension, if the Yamabe invariant of connected sum of two copies of $M$ attains its minimum, then the same is true for $L^\infty$ metrics with isolated point singularities., Comment: 27 pages

The deformed Hermitian-Yang-Mills equation is a complex Hessian equation on compact K\"ahler manifolds that corresponds to the special Lagrangian equation in the context of the Strominger-Yau-Zaslow mirror symmetry. Recently, Chen proved that the existence of the solution is equivalent to a uniform stability condition in terms of holomorphic intersection numbers along test families. In this paper, we establish an analogous stability result not involving a uniform constant in accordance with a recent work on the $J$-equation by Song, which makes further progress toward Collins-Jacob-Yau's original conjecture in the supercritical phase case. In particular, we confirm this conjecture for projective manifolds in the supercritical phase case., Comment: 43 pages, added new references and Corollary 1.5 on Collins-Jacob-Yau's conjecture for projective manifolds

In this work, we study the H\"older regularity of the K\"ahler- Ricci flow on compact K\"ahler manifolds with semi-ample canonical line bundle. By adapting the method in the work of Hein-Tosatti on collapsing Calabi-Yau metrics, we obtain a uniform spatial H\"older estimate of the K\"ahler-Ricci flow for all time., Comment: 15 pages

In this work, we consider sequence of metrics with almost non-negative scalar curvature on torus. We show that if the sequence is uniformly conformal to another sequence of metrics with uniformly controlled geometry, then it converges to a flat metric in the volume preserving intrinsic flat sense, $L^{p}$ sense and the measured Gromov-Hausdorff sense., Comment: 21 pages, minor mistakes fixed

We show that a complete Ricci flow of bounded curvature which begins from a manifold with a Ricci lower bound, local entropy bound, and small local scale-invariant integral curvature control will have global point-wise curvature control at positive times. As applications, we obtain under similar assumptions a compactness result and a gap theorem for complete noncompact manifolds with nonnegative Ricci Curvature., Comment: 29 pages, reference updated, result added, minor mistakes fixed

Consider a sequence of Riemannian manifolds $(M^n_i,g_i)$ with scalar curvatures and entropies bounded below by small constants $R_i,\mu_i \geq-\epsilon_i$. The goal of this paper is to understand notions of convergence and the structure of limits for such spaces. Even in the seemingly rigid case $\epsilon_i\to 0$, we construct examples showing that such a sequence may converge wildly in the Gromov-Hausdorff or Intrinsic Flat sense. On the other hand, we will see that these classical notions of convergence are the incorrect ones to consider. Indeed, even a metric space is the wrong underlying category to be working on. Instead, we introduce $d_p$ convergence, a weaker notion of convergence that is valid for a class of rectifiable Riemannian spaces. These rectifiable spaces have well-behaved topology, measure theory, and analysis, though potentially there will be no reasonably associated distance function. Under the $d_p$ notion of closeness, a space with almost nonnegative scalar curvature and small entropy bounds must in fact be close to Euclidean space; this will constitute our $\epsilon$-regularity theorem. More generally, we have a compactness theorem saying that sequences of Riemannian manifolds $(M^n_i,g_i)$ with small lower scalar curvature and entropy bounds $R_i,\mu_i \geq -\epsilon$ must $d_p$ converge to such a rectifiable Riemannian space $X$. Comparing to the first paragraph, the distance functions of $M_i$ may be degenerating, even though in a well-defined sense the analysis cannot be. Applications for manifolds with small scalar and entropy lower bounds include an $L^\infty$-Sobolev embedding and apriori $L^p$ scalar curvature bounds for $p<1$., Comment: added reference and result of torus stability

In this work we consider compact K\"ahler manifolds with non-positive mixed curvature which is a "convex combination" of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef. Moreover, if the curvature is negative at some point, then the manifold is projective with canonical line bundle being big and nef. If in addition the curvature is negative, then the canonical line bundle is ample. As an application, we answer a question of Ni concerning manifolds with negative $k$-Ricci curvature and generalize a result of Wu-Yau and Diverio-Trapani to the conformally K\"ahler case. We also show that the compact K\"ahler manifold is projective and simply connected if the mixed curvature is positive., Comment: 22 pages, final version

In this work, we study the deformation of Hermitian metrics with Chern connection. By adapting the conformal perturbation method of Aubin and Ehrlich to Hermitian setting, we prove that Hermitian metrics with quasi-positive (resp. quasi-negative) second Chern-Ricci curvature can be deformed to one with positive (resp. negative) curvature., Comment: 10 pages; references added

In this note, we establish a local maximum principle along Ricci flow under scaling invariant curvature condition. This unifies the known preservation of nonnegativity results along Ricci flow with unbounded curvature. By combining with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard's localized version of a maximum principle given by R. Bamler, E. Cabezas-Rivas, and B. Wilking on the lower bound of curvature conditions., Comment: 25 pages

The space $\mathcal{H}$ of "almost calibrated" $(1,1)$ forms on a compact K\"ahler manifold plays an important role in the study of the deformed Hermitian-Yang-Mills equation of mirror symmetry as emphasized by recent work of the second author and Yau, and is related by mirror symmetry to the space of positive Lagrangians studied by Solomon. This paper initiates the study of the geometry of $\mathcal{H}$. We show that $\mathcal{H}$ is an infinite dimensional Riemannian manifold with non-positive sectional curvature. In the hypercritical phase case we show that $\mathcal{H}$ has a well-defined metric structure, and that its completion is a ${\rm CAT}(0)$ geodesic metric space, and hence has an intrinsically defined ideal boundary. Finally, we show that in the hypercritical phase case $\mathcal{H}$ admits $C^{1,1}$ geodesics, improving a result of the second author and Yau. Using results of Darvas-Lempert we show that this result is sharp., Comment: 50 pages

In this article, we study the higher-order regularity of the K\"ahler-Ricci flow on compact K\"ahler manifolds with semi-ample canonical line bundle. We proved, using a parabolic analogue of Hein-Tosatti's work on collapsing Calabi-Yau metrics, that when the generic fibers of the Iitaka fibration are biholomorphic to each other, the flow converges locally smoothly away from singular fibers to a negative K\"ahler-Einstein metric on the base manifold. In particular, we proved that the Ricci curvature of the flow is uniformly bounded on any compact subsets away from singular fibers when the generic fibers are biholomorphic to each other., Comment: 30 pages

We prove some uniqueness result for solutions to the heat equation on Riemannian manifolds. In particular, we prove the uniqueness of $L^p$ solutions with $0< p< 1$, and improves the $L^1$ uniqueness result of P. Li by weakening the curvature assumption., Comment: 7 pages

We show the existence of complete negative K\"ahler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature. We prove that any K\"ahler metrics on such manifolds can be deformed to the complete negative K\"ahler-Einstein metric using the normalized K\"ahler-Ricci flow.

In this paper, we construct local and global solutions to the K\"ahler-Ricci flow from a non-collapsed K\"ahler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed K\"ahler manifolds with orthogonal bisectional curvature and Ricci curvature bounded from below is homeomorphic to a complex manifold. We also use it to study the complex structure of complete K\"ahler manifolds with nonnegative orthogonal bisectional curvature, nonnegative Ricci curvature and maximal volume growth., Comment: 32 pages

We produce complete bounded curvature solutions to K\"ahler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth \K metric uniformly equivalent to another complete bounded curvature \K metric. We obtain related flow results for non-smooth as well as degenerate initial conditions. We also obtain a stability result for complex space forms under the flow., Comment: 22 pages; editions made to earlier version (including revision of title, addition to Theorem 1.2 and correction in proof of Claim 2.2)

We prove that compact complex manifolds with admitting metrics with negative Chern curvature operator either admit a $dd^c$-exact positive (1,1) current, or are K\"ahler with ample canonical bundle. In the case of complex surfaces we obtain a complete classification. The proofs rely on a global existence and convergence result for the pluriclosed flow.

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 for surfaces of hyperbolic type of Ricci flow to higher dimensions in certain sense. On the other hand, we also discuss the long time behaviour of the solution and obtain some sufficient conditions for the existence of K\"ahler-Einstein metric on complete noncompact Hermitian manifolds, which generalizes the work of Lott-Zhang and Tosatti-Weinkove to complete noncompact Hermitian manifolds with possibly unbounded curvature., Comment: 40 pages

In this work, we first establish short time existence and Shi's type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of Kaehler-Einstein metric on complete noncompact Hermitian manifolds., Comment: 28 pages

In this article we use Ricci flow to show that complete PIC1 manifolds with maximal volume growth are diffeomorphic to $\mathbb{R}^n$. One of the key ingredients is local estimates of curvature lower bounds on an initial time interval of the Ricci flow. As another application of these estimates we obtain pseudolocality type results related to the PIC1 condition., Comment: 18 pages