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We derive some estimates for stable minimal hypersurfaces in $R^{n+1}$. The estimates are related to recent proofs of Bernstein theorems for complete stable minimal hypersurfaces in $R^{n+1}$ for $3\le n\le 5$ by Chodosh-Li, Chodosh-Li-Minter-Stryker and Mazet. In particular, the estimates indicate that the methods in their proofs may not work for $n=6$, which is observed also by Antonelli-Xu and Mazet. The method of derivation in this work might also be applied to other problems., Comment: References have been updated

Motivated by previous study on mean curvature flow and prescribed mean curvature flow on spatially compact space or asymptotically flat spacetime, in this work we will find sufficient conditions for the short time existence of prescribed mean curvature flow on a Lorentz manifold with a smooth time function starting from a complete noncompact spacelike hypersurface. Long time existence and convergence will also be discussed. Results will be applied to study some prescribed mean curvature flows inside the future of the origin in the Minkowski spacetime. Examples of spacetime related to the existence and convergence results near the future null infinity of the Schwarzschild spacetime are also discussed.

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

Motivated by a result of Treibergs, given a smooth function f(y) on the standard sphere S^2, and any positive constant H_0, we construct a spacelike surface with constant mean curvature H_0 in the Schwarzschild spacetime, which is the graph of a function u(y, r) defined on r>r_0 for some r_0>0 in the standard coordinates exterior to the blackhole. Moreover, u has the following asymptotic behavior: |u(y,r)-r_*-(f(y)+r^{-1}\phi(y)+1/2 r^{-2}\psi(y)|\le Cr^{-3} for some C>0, where r_*=r+2m\log(r/(2m)-1). Here \phi, \psi are functions determined by f and H_0. In particular, the surface intersects the future null infinity with the cut given by the function f. In addition, we prove that the function u-r_* is uniformly Lipschitz near the future null infinity., Comment: Lipschitzian regularity of the solution has been added

We derive local and global monotonic quantities associated to $p$-harmonic functions on manifolds with nonnegative scalar curvature. As applications, we obtain inequalities relating the mass of asymptotically flat $3$-manifolds, the $p$-capacity and the Willmore functional of the boundary. As $ p \to 1$, one of the results retrieves a classic relation that the ADM mass dominates the Hawking mass if the surface is area outer-minimizing., Comment: 38 pages, to appear in PAMQ

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

We prove that a spacelike spherical symmetric constant mean curvature (SSCMC) surface and a general spacelike constant mean curvature (CMC) surface with certain boundary condition at the future null-infinity in Schwarzschild spacetime are asymptotically hyperbolic in the sense of Wang \cite{Wang2001} and Chru\'sciel-Herzlich \cite{ChruscielHerzlich} respectively. Near the future null-infinity ($s=0$), we derive that the boundary data of spacelike CMC surfaces can be expressed as those on $\mathbb{S}^{2}$ up to three order and obtain a compatibility condition for fourth order derivatives near $s=0$. We also show that if the trace free part of the second fundamental forms $\mathring A$ of this spacelike CMC surface decay fast enough then the restriction of its associate function $P$ (for definition, see \eqref{defofp} ) on the null-infinity must be a first eigenfunction of the Laplace on $\mathbb{S}^2$ or constant. In particular in Minkowski spacetime, a uniqueness result and constructions of spacelike CMC surfaces near $s=0$ are proved. Also, we show that the inner boundary of certain spacelike CMC surfaces are totally geodesic., Comment: 41 pages, all comments are welcome

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 obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by Li-Tam [The heat equation and harmonic maps of complete manifolds, Invent. Math., 1991] and Chen-Zhu [Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geometry, 2006]. In particular, we prove the short time existence of harmonic map heat flow along a complete Ricci flow $g(t)$ on $M$ into a complete manifold with curvature bounded from above with a smooth initial map of uniformly bounded energy density, under the assumptions that $|\text{Rm}(g(t))|\leq a/t$ and $g(t)$ is uniformly equivalent to $g(0)$., Comment: 29 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

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 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

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

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 short note, we prove positivity of Brown-York mass under quasi-positive boundary data which generalize some previous results by the authors. The corresponding rigidity result is obtained., Comment: 15 pages, all comments are welcome

In this work, we obtain a existence criteria for the longtime K\"ahler Ricci flow solution. Using the existence result, we generalize a result by Wu-Yau on the existence of K\"ahler Einstein metric to the case with possibly unbounded curvature. Moreover, the K\"ahler Einstein metric with negative scalar curvture must be unique up to scaling., Comment: 25 pages

We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown--York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them., Comment: 46 pages. Latest pre-print version. Final and most up-to-date version will appear in Journal fur die reine und angewandte Mathematik (Crelle)

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 any complete noncollapsed Kahler metric with nonnegative bisectional curvature on a noncompact complex manifold can be deformed to a complete Kahler metric with nonnegative and bounded bisectional curvature which will have maximal volume growth if the initial metric has maximal volume. Combining this result with the result of Chau-Tam, we give another proof that a complete noncompact Kahler manifold with nonnegative bisectional curvature (not necessarily bounded) and maximal volume growth is biholomorphic to the complex Euclidean space. This last result has already been proved by Gang Liu recently using other methods. This last result is partial confirmation of a uniformization conjecture of Yau., Comment: some results added; printing mistakes corrected

In this work, using the method by He, we prove a short time existence for Ricci flow on a complete noncompact Riemannian manifold with the following properties: (i) there is $r_0>0$ such that the volume of any geodesic balls of radius $r\le r_0$ is close to the volume of the geodesic ball in the Euclidean space with the same radius; (ii) the Ricci curvature is bounded below; (iii) the sectional curvature is bounded below by $-Cd^2(x,x_0)$ where $d(x,x_0)$ is the distance from a fixed point. An application to the uniformization for complete noncompact Kahler manifolds with nonnegative bisectional curvature and maximal volume growth is given. The result is related to the results of He and Liu. We prove if $g(t)$ is a complete solution to the Kahler-Ricci flow on a complete noncompact Kahler manifold so that $g(0)$ has nonnegative bisectional curvature and the curvature of $g(t)$ is bounded by $at^{-b}$ for some $a>0, 2>b>0$, then $g(t)$ also has nonnegative bisectional curvature. This generalizes a previous result. Under the assumption that $g(0)$ is noncollapsing, with additional assumption that the curvature of $g(t)$ is uniformly bounded away from $t>0$, we prove that the curvature of $g(t)$ is in fact bounded by $at^{-1}$ for some $a>0$. This result is proved using the method by Simon and Topping., Comment: This is an expanded version of the previous paper "A note on existence of Ricci flow"

Let $M^n$, $n\ge3$, be a compact differentiable manifold with nonpositive Yamabe invariant $\sigma(M)$. Suppose $g_0$ is a continuous metric with $V(M, g_0)=1$, smooth outside a compact set $\Sigma$, and is in $W^{1,p}_{loc}$ for some $p>n$. Suppose the scalar curvature of $g_0$ is at least $\sigma(M)$ outside $\Sigma$. We prove that $g_0$ is Einstein outside $\Sigma$ if the codimension of $\Sigma$ is at least $2$. If in addition, $g_0$ is Lipschitz then $g_0$ is smooth and Einstein after a change the smooth structure. If $\Sigma$ is a compact embedded hypersurface, and $g_0$ is smooth up to $\Sigma$ from two sides of $\Sigma$, and if the difference of the mean curvatures along $\Sigma$ at two sides of $\Sigma$ has a fixed appropriate sign. Then $g_0$ is also Einstein outside $\Sigma$. For manifolds with dimension between $3$ and $7$ without spin assumption, we obtain a positive mass theorem on an asymptotically flat manifold for metrics with a compact singular set of codimension at least $2$., Comment: 47pages, All comments are welcome

W. Simon proved a conformal positive mass theorem, which was used to prove uniqueness of black holes later. In this note, we will generalize Simon's conformal positive mass theorem in two directions. First we will consider spacetime version of conformal positive mass theorems on asymptotically flat initial data set. Next, we will prove a conformal positive mass theorem on asymptotically hyperbolic manifolds., Comment: some printing mistakes in the statements of Theorem 1.1 and Theorem 2.1 are corrected

We analyze critical points of two functionals of Riemannian metrics on compact manifolds with boundary. These functionals are motivated by formulae of the mass functionals of asymptotically flat and asymptotically hyperbolic manifolds.

In this work we study the quasilocal energy as in [11] for a constant radius surface in Kerr spacetime in Boyer-Lindquist coordinates. We show that under suitable conditions for isometric embedding, for a stationary observer the quasilocal energy defined in [11] for constant radius in a Kerr like spacetime is exactly equal to the Brown-York quasilocal energy [2]. By some careful estimations, we show that for a constant radius surface in the Kerr spacetime which is outside the ergosphere the embedding conditions for the previous result are satisfied. We discuss extremal solutions as described in [14]. We prove a uniqueness result. We find all extremal solutions in the Minkowski spacetime. Finally, we show that near the horizon of the Kerr spacetime for the small rotation case the extremal solutions are trivial., Comment: There are some contents added in section 4 and also some refinement on abstract and introduction

On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown-York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions., Comment: References updated, introduction revised

Fix a complete noncompact \K manifold $(M^n,h_0)$ with bounded curvature. Let $g(t)$ be a bounded curvature solution to the \KR flow starting from some $g_0$ uniformly equivalent to $h_0$. We estimate the existence time of $g(t)$ together with $C^0$ bounds and curvature bounds, where the estimates depend only on $h_0$ and the $C^0$ distance between $g_0$ and $h_0$. We also generalize these results to cases when $g_0$ may have unbounded curvature., Comment: 12 pages, minor corrections, proof of Theorem 2.3 modified, references updated

Let $g(t)$ be a complete solution to the Ricci flow on a noncompact manifold such that $g(0)$ is Kahler. We prove that if $|Rm(g(t))|_{g(t)}\le a/t$ for some $a>0$, then $g(t)$ is Kahler for $t>0$. We prove that there is a constant $ a(n)>0$ depending only on $n$ such that the following is true: Suppose $g(t)$ is a complete solution to the Kahler-Ricci flow on a noncompact $n$-dimensional complex manifold such that $g(0)$ has nonnegative holomorphic bisectional curvature and such that $|Rm(g(t))|_{g(t)}\le a(n)/t$, then $g(t)$ has nonnegative holomorphic bisectional curvature for $t>0$. These generalize the results by W. S. Shi. As corollaries, we prove that (i) any complete noncompact Kahler manifold with nonnegative complex sectional curvature with maximum volume growth is biholomorphic to $ C^n$; and (ii) there is $\epsilon(n)>0$ depending only on $n$ such that if $(M^n,g_0)$ is a complete noncompact Kahler manifold of complex dimension $n$ with nonnegative holomorphic bisectional curvature and maximum volume growth and if $(1+\epsilon(n))^{-1}h\le g_0\le (1+\epsilon(n))h$ for some Riemannian metric $h$ with bounded curvature, then $M$ is biholomorphic to $C^n$.

We produce longtime solutions to the K\"ahler-Ricci flow for complete K\"ahler metrics on $\Bbb C ^n$ without assuming the initial metric has bounded curvature, thus extending results in [3]. We prove the existence of a longtime bounded curvature solution emerging from any complete $U(n)$-invariant K\"ahler metric with non-negative holomorphic bisectional curvature, and that the solution converges as $t\to \infty$ to the standard Euclidean metric after rescaling. We also prove longtime existence results for more general K\"ahler metrics on $\Bbb C ^n$ which are not necessarily $U(n)$-invariant., Comment: 27 pages; condition (c2) modified in section 3; correction to statement (part (b)) and proof of Proposition 3.1; reference [11] added; typos corrected

We prove directly without using a density theorem that (i) the ADM mass defined in the usual way on an asymptotically flat manifold is equal to the mass defined intrinsically using Ricci tensor; (ii) the Hamiltonian formulation of center of mass and the center of mass defined intrinsically using Ricci tensor are the same., Comment: update references

We consider the question whether a static potential on an asymptotically flat 3-manifold can have nonempty zero set which extends to the infinity. We prove that this does not occur if the metric is asymptotically Schwarzschild with nonzero mass. If the asymptotic assumption is relaxed to the usual assumption under which the total mass is defined, we prove that the static potential is unique up to scaling unless the manifold is flat. We also provide some discussion concerning the rigidity of complete asymptotically flat 3-manifolds without boundary that admit a static potential., Comment: introduction revised; an outline of a space-time approach added

We produce solutions to the K\"ahler-Ricci flow emerging from complete initial metrics $g_0$ which are $C^0$ Hermitian limits of K\"ahler metrics. Of particular interest is when $g_0$ is K\"ahler with unbounded curvature. We provide such solutions for a wide class of $U(n)$-invariant K\"ahler metrics $g_0$ on $n$ dimensional complex Euclidean space, many of which having unbounded curvature. As a special case we have the following Corollary: The K\"ahler-Ricci flow has a smooth short time solution starting from any smooth complete $U(n)$-invariant K\"abler metric on $\C^n$ with either non-negative or non-positive holomorphic bisectional curvature, and the solution exists for all time in the case of non-positive curvature., Comment: 32 pages; v2 minor corrections and editions, reference added, Corollary 4.3 removed due to mistake in proof, precise dependence of $T$ on initial constants removed from statement of Theorem 5.4

We will discuss existence of center of mass on asymptotically Schwarzschild manifold defined by Huisken-Yau and Corvino-Schoen. Conditions of existence and examples on non existence are given.

We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1. The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen. We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We also obtain some partial results on the comparison of the normalized Stekov eigenvalues of rotationally symmetric metrics and general conformal metrics on the cylinder. A counter example is constructed to show that for that the first normalized Steklov eigenvalue of rotationally symmetric metric may not be larger., Comment: 26 pages

Motivated by the quasi-local mass problem in general relativity, we apply the asymptotically flat extensions, constructed by Shi and Tam in the proof of the positivity of the Brown--York mass, to study a fill-in problem of realizing geometric data on a 2-sphere as the boundary of a compact 3-manifold of nonnegative scalar curvature. We characterize the relationship between two borderline cases: one in which the Shi--Tam extension has zero total mass, and another in which fill-ins of nonnegative scalar curvature fail to exist. Additionally, we prove a type of positive mass theorem in the latter case., Comment: 14 pages

We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang-Yau quasi-local energy in general relativity. We prove that the functional is positive definite on large coordinate spheres, and more general on nearly round surfaces including large constant mean curvature spheres in asymptotically flat 3-manifolds with positive mass; it is also positive definite on small geodesics spheres, whose centers do not have vanishing curvature, in Riemannian 3-manifolds of nonnegative scalar curvature. We also give examples of functions H, which can be made arbitrarily close to the constant 2, on the standard sphere such that the boundary data consisting of the standard spherical metric and H has positive Brown-York mass while the associated functional is negative somewhere., Comment: 36 pages

In this article we give necessary and sufficient conditions for an irreducible K\"ahler C-space with $b_2=1$ to have nonnegative or positive quadratic bisectional curvature, assuming the space is not Hermitian symmetric. In the cases of the five exceptional Lie groups $E_6, E_7, E_8, F_4, G_2$, the computer package MAPLE is used to assist our calculations. The results are related to two conjectures of Li-Wu-Zheng., Comment: 64 pages. Minor editions and corrections made to previous version

In this paper, we develop a method of solving the Poincar\'e-Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge-Laplace heat equation on $(1, 1)$-forms. The method is effective in proving an optimal result when $M$ has nonnegative bisectional curvature. It also provides an alternate proof of a recent gap theorem of the first author.

In this paper, we provide some remarks on the scalar curvature rigidity theorem of Brendle and Marques in \cite{BrendleMarques}. The main result is that Brendle and Marques' theorem holds on a geodesic ball larger than that specified in [2]., Comment: the main result strengthened

Motivated by Brendle-Marques-Neves' counterexample to the Min-Oo's conjecture, we prove a volume constrained scalar curvature rigidity theorem which applies to the hemisphere., Comment: Theorem 1.7 and Theorem 5.1 strengthened

This is a continuous work about the nonexistence of some complete metrics on the product of two manifolds studied by Tam-Yu [Asian Journal of Mathematics, 14(2010)]. Motivated by the result of Tossati [Comm.Anal.Geom. 15(2007)]. We generalize the corresponding results of Tam-Yu [Asian Journal of Mathematics, 14(2010)] to the almost-Hermitian case.

In this article we study compact K\ahler manifolds satisfying a certain nonnegativity condition on the bisectional curvature. Under this condition, we show that the scalar curvature is nonnegative and that the first Chern class is positive assuming local irreducibility. We also obtain a partial classification of possible de Rham decompositions of the universal cover under this condition, Comment: 14 pages

Motivated by the results of Wu-Yau-Zheng \cite{WuYauZheng}, we show that under a certain curvature assumption the harmonic representative of any boundary class of the K\"ahler cone is nonnegative., Comment: v2: Footnote added on page 1. Accompanying reference also added

In this paper, we will show that the limit of some quasilocal mass integrals of the coordinate spheres in an asymptotically hyperbolic (AH) manifold is the mass integral of the AH manifold. This is the analogue of the well known result that the limit of the Brown-York mass of coordinate spheres is the ADM mass in an asymptotically flat manifold., Comment: 16 pages

In this work, we will verify some comparison results on Kahler manifolds. They are complex Hessian comparison for the distance function from a closed complex submanifold of a Kahler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scalar curvature. This work is motivated by comparison results of Li and Wang ., Comment: 14 pages

In this paper, we prove the following theorem regarding the Wang-Yau quasi-local energy of a spacelike two-surface in a spacetime: Let $\Sigma$ be a boundary component of some compact, time-symmetric, spacelike hypersurface $\Omega$ in a time-oriented spacetime $N$ satisfying the dominant energy condition. Suppose the induced metric on $\Sigma$ has positive Gaussian curvature and all boundary components of $\Omega$ have positive mean curvature. Suppose $H \le H_0$ where $H$ is the mean curvature of $\Sigma$ in $\Omega$ and $H_0$ is the mean curvature of $\Sigma$ when isometrically embedded in $R^3$. If $\Omega$ is not isometric to a domain in $R^3$, then 1. the Brown-York mass of $\Sigma$ in $\Omega$ is a strict local minimum of the Wang-Yau quasi-local energy of $\Sigma$, 2. on a small perturbation $\tilde{\Sigma}$ of $\Sigma$ in $N$, there exists a critical point of the Wang-Yau quasi-local energy of $\tilde{\Sigma}$., Comment: substantially revised, main theorem replaced, Section 3 added

We study a parabolic complex Monge-Amp\`{e}re type equation of the form \eqref{MA} on a complete noncompact \K manifold. We prove a short time existence result and obtain basic estimates. Applying these results, we prove that under certain assumptions on a given real and closed (1,1) form $\Omega$ and initial \K metric $g_0$ on $M$, the modified \KR flow $g'=-\Ric+\Omega$ has a long time smooth solution converging to a complete \K metric such that $\Ric=\Omega$, which extends the result in [1] to non-compact manifolds. We will also obtain a long time existence result for the \KR flow which generalizes a result [5]., Comment: 30 pages

Let $M=X\times Y$ be the product of two complex manifolds of positive dimensions. In this paper, we prove that there is no complete K\"ahler metric $g$ on $M$ such that: either (i) the holomorphic bisectional curvature of $g$ is bounded by a negative constant and the Ricci curvature is bounded below by $-C(1+r^2)$ where $r$ is the distance from a fixed point; or (ii) $g$ has nonpositive sectional curvature and the holomorphic bisectional curvature is bounded above by $-B(1+r^2)^{-\delta}$ and the Ricci curvature is bounded below by $-A(1+r^2)^\gamma$ where $A, B, \gamma, \delta$ are positive constants with $\gamma+2\delta<1$. These are generalizations of some previous results, in particular the result of Seshadri and Zheng., Comment: 11 pages