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Firstly, we confirm a conjecture asserting that any compact Kähler manifold N with Ric ⊥ > 0 {\operatorname{Ric}^{\perp}>0} must be simply-connected by applying a new viscosity consideration to Whitney’s comass of ( p , 0 ) {(p,0)} -forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension n under the condition Ric k > 0 {\operatorname{Ric}_{k}>0} (for some k ∈ { 1 , … , n } {k\in\{1,\dots,n\}} , with Ric n {\operatorname{Ric}_{n}} being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollár, Miyaoka and Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a second variation consideration of [L. Ni and F. Zheng, Positivity and Kodaira embedding theorem, preprint 2020, https://arxiv.org/abs/1804.09696]. Thirdly, motivated by Ric ⊥ {\operatorname{Ric}^{\perp}} and the classical work of Calabi and Vesentini [E. Calabi and E. Vesentini, On compact, locally symmetric Kähler manifolds, Ann. of Math. (2) 71 1960, 472–507], we propose two new curvature notions. The cohomology vanishing H q ⁢ ( N , T ′ ⁢ N ) = { 0 } {H^{q}(N,T^{\prime}N)=\{0\}} for any 1 ≤ q ≤ n {1\leq q\leq n} and a deformation rigidity result are obtained under these new curvature conditions. In particular, they are verified for all classical Kähler C-spaces with b 2 = 1 {b_{2}=1} . The new conditions provide viable candidates for a curvature characterization of homogeneous Kähler manifolds related to a generalized Hartshone conjecture.

Unmanned aerial vehicle- (UAV-) assisted communication has great potential to provide on-demand wireless services and improve the outdoor link throughput. In this paper, a UAV-based cognitive radio network (CRN) is investigated in which the UAV works as a secondary user (SU). Considering the overlay spectrum sensing mode, the UAV can operate on the licensed spectrum bands of primary user (PU) only when PU is idle. In each working frame structure, both sensing time slot and transmission time slot are analysed in radians. Specifically, our objective is to maximize the spectrum efficiency (SE) of the UAV by jointly optimizing the sensing radian and the number of radians. For the single-radian and multiradian schemes, the dichotomy and alternative iterative optimization (AIO) algorithm are proposed to solve the SE optimization problem. Simulation results show that the proposed multiradian cooperative spectrum sensing (CSS) scheme can achieve better performance on ensuring the quality-of-service (QoS) of the PU, and it can significantly enhance the SE of the UAV especially in the severe channel environments.

In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature upper bound. New results on ancient solutions for the Ricci and Kahler-Ricci flow are obtained. Applications to Kahler manifolds with almost nonnegative orthogonal bisectional curvature are derived as consequences. The main new feature is that no curvature upper bound is assumed.

The auto-ignition temperature (AIT) is one of the most important parameters in flammability risk assessment and management in the chemical process. Therefore, in this work, quantitative structure–property relationship approach was employed to estimate the AIT of binary liquid mixtures only based on the information of molecular structures. Various kinds of molecular descriptors were calculated using Dragon 6.0 software after the geometry optimization of molecular structures. Genetic algorithm (GA) was used to select the best subset of descriptors which have a significant contribution to AIT. Two novel models including multiple linear regression (MLR) model and support vector machine (SVM) model were developed based on the GA-selected molecular descriptors. The resulted models showed satisfied goodness-of-fit, robustness and external predictability after the rigorous verification based on appropriate criteria. The MLR model showed great performance with the average absolute error (AAE) of training set and test set being 13.420 °C and 15.076 °C, while the AAE of SVM model was reduced to 5.629 °C and 9.206 °C, respectively. The two optimal models could provide a convenient and effective way to predict the AIT of binary liquid mixtures as well as guidance for the safety design of the chemical process industry.

In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author [L. Ni, An optimal gap theorem, Invent. Math. 189 2012, 3, 737–761]. We also prove a Liouville theorem for plurisubharmonic functions on such a manifold, which generalizes a previous result of L.-F. Tam and the first author [L. Ni and L.-F. Tam, Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature, J. Differential Geom. 64 2003, 3, 457–524] and complements a recent result of Liu [G. Liu, Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds, Duke Math. J. 165 2016, 15, 2899–2919].

By studying a complex Monge–Ampère equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact Kähler manifold [Formula: see text] with [Formula: see text] for some integer [Formula: see text] with [Formula: see text], and the ampleness of the canonical line bundle [Formula: see text].

We prove estimates interpolating the Schwarz Lemmata of Royden–Yau and the ones recently established by the author. These more flexible estimates provide additional information on (algebraic) geometric aspects of compact Kähler manifolds with nonnegative holomorphic sectional curvature, nonnegative [Formula: see text] or positive [Formula: see text].

In this paper, we consider orthogonal Ricci curvature $$Ric^{\perp }$$ for Kahler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate subtle relationship among them. As a consequence of the vanishing theorem, we show that any compact Kahler manifold with positive orthogonal Ricci curvature must be projective. This result complements a recent result of Yang (RC-positivity, rational connectedness, and Yau’s conjecture. arXiv:1708.06713 ) on the projectivity under the positivity of holomorphic sectional curvature. The simply-connectedness is shown when the complex dimension is smaller than five. Further study of compact Kahler manifolds with $$Ric^{\perp }>0$$ is carried in Ni et al. (Manifolds with positive orthogonal Ricci curvature. arXiv:1806.10233 ).

A spreading system based on weighted fractional Fourier transform (WFRFT) is proposed over the channel with additive white Gaussian noise (AWGN) and narrowband interference (NBI). The cross-correlation coefficient between pseudo-noise (PN) sequence and NBI (CPI) and the proposed system's bit error rate (BER) are deduced. We proved that the optimal WFRFT order can be obtained to achieve the best BER performance through minimization of CPI. Numerical results show that the optimal WFRFT order can reduce the adverse effect caused by NBI in different distribution states.

In this paper we study the class of compact Kähler manifolds with positive orthogonal Ricci curvature: $Ric^\perp>0$. First we illustrate examples of Kähler manifolds with $Ric^\perp>0$ on Kähler C-spaces, and construct ones on certain projectivized vector bundles. These examples show the abundance of Kähler manifolds which admit metrics of $Ric^\perp>0$. Secondly we prove some (algebraic) geometric consequences of the condition $Ric^\perp>0$ to illustrate that the condition is also quite restrictive. Finally this last point is made evident with a classification result in dimension three and a partial classification in dimension four.

In this paper we prove uniform regularity estimates for the normalized Gauss curvature flow in higher dimensions. The convergence of solutions in $C^\infty$-topology to a smooth strictly convex soliton as $t$ approaches to infinity is obtained as a consequence of these estimates together with an earlier result of Andrews. The estimates are established via the study of a new entropy functional for the flow.

We prove that convex hypersurfaces in R n + 1 contracting under the flow by any power α > 1 n + 2 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the initial body we prove that the limit is the round sphere for α ≥ 1 .

In order to assess the oxidation self-heating hazard of sulfurized rust, for particular ambient conditions in crude oil tanks, the support vector machine (SVM) technique is applied to predict the maximum temperature ( T max ) of oxidation self-heating process. Five governing parameters are selected, i.e. the water content, mass of sulfurized rust, operating temperature, air flow rate and oxygen concentration in the respiratory/safety valve. The efficiency and validity of the SVM predictions are investigated in the case of two sets of data: more than 85 experiments performed in academic lab (China) and almost 17 additional results collected from existing literature. Two main steps are also discussed: the training process (on selected subsets of data) and prediction process (for the remaining subsets of data). It can be concluded that for both datasets the maximum temperature ( T max ) values calculated by SVM technique were in good accordance with the experimental results, with relative errors smaller than 15% except for a few cases. The SVM technique seems therefore to be relevant and very helpful for complex implicit processes such as chemical reactions, as it is the case of the oxidation of sulfurized rust in oil tanks. Furthermore, such predictive methods can be continuously be improved through additional experiments feedback (larger databases) and can then be of crucial help for monitoring and early warning of hazardous reactions.

Due to the intense vibration during launching and rigorous orbital temperature environment, the kinematic parameters of space robot may be largely deviated from their nominal parameters. The disparity will cause the real pose (including position and orientation) of the end effector not to match the desired one, and further hinder the space robot from performing the scheduled mission. To improve pose accuracy of space robot, a new self-calibration method using the distance measurement provided by a laser-ranger fixed on the end-effector is proposed. A distance-measurement model of the space robot is built according to the distance from the starting point of the laser beam to the intersection point at the declining plane. Based on the model, the cost function about the pose error is derived. The kinematic calibration is transferred to a non-linear system optimization problem, which is solved by the improved differential evolution (DE) algorithm. A six-degree of freedom (6-DOF) robot is used as a practical simulation example, and the simulation results show: 1) A significant improvement of pose accuracy of space robot can be obtained by distance measurement only; 2) Search efficiency is increased by improved DE; 3) More calibration configurations may make calibration results better.

In this paper we study the heat equation (of Hodge Laplacian) deformation of .p;p/-forms on a Kahler manifold. After identifying the condition and establishing that the positivity of a .p;p/-form solution is preserved under such an invariant condition, we prove the sharp differential Harnack (in the sense of LiYau-Hamilton) estimates for the positive solutions of the Hodge Laplacian heat equation. We also prove a nonlinear version coupled with the Kahler-Ricci flow and some interpolating matrix differential Harnack-type estimates for both the Kahler-Ricci flow and the Ricci flow. © 2011 Wiley Periodicals, Inc.

We say that a nonnegatively curved manifold $(M,g)$ has quarter pinched flag curvature if for any two planes which intersect in a line the ratio of their sectional curvature is bounded above by 4. We show that these manifolds have nonnegative complex sectional curvature. By combining with a theorem of Brendle and Schoen it follows that any positively curved manifold with strictly quarter pinched flag curvature must be a space form. This in turn generalizes a result of Andrews and Nguyen in dimension 4. For odd dimensional manifolds we obtain results for the case that the flag curvature is pinched with some constant below one quarter, one of which generalizes a recent work of Petersen and Tao.

In this work we derive local gradient and Laplacian estimates of the Aronson–Benilan and Li–Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discover some new entropy formulae for these equations.

We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding solitons with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. The geometric invariant appearing in the sharp lower bound is shown to be nonnegative. We also characterize the expanders when such invariant is zero. In the proof, various useful volume growth estimates are also established for gradient shrinking and expanding solitons. In particular, we prove that the asymptotic volume ratio of any gradient shrinking soliton with nonnegative Ricci curvature must be zero.

The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. Moreover in dimension three our proof generalizes Perelman's result by removing the $\kappa$-non-collapsing assumption and allowing general curvature growth. The method also allows us to prove a classification result on gradient shrinking solitons with vanishing Weyl curvature tensor in high dimensions, which includes the rotationally symmetric ones.

This is a survey article for Surveys in Differential Geometry series on the subject of Li-Yau-Hamilton type differential inequalities and related monotonicity formulae.

In this short note, as a simple application of the strong result proved recently by Böhm and Wilking, we give a classification on closed manifolds with 2 2 -nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifold (with dimension ≥ 3 \ge 3 ) whose curvature operator is bounded and satisfies the pinching condition R ≥ δ tr ⁡ ( R ) 2 n ( n − 1 ) I > 0 R\ge \delta \frac {\operatorname {tr}(R)}{2n(n-1)} \mathrm {I}>0 , for some δ > 0 \delta >0 , must be compact. This provides an intrinsic analogue of a result of Hamilton on convex hypersurfaces.

We provide a proof for a Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators based on the strong maximum principle. This proof is modeled after a variational proof of...

In [L-Y], Peter Li and S.-T. Yau developed the fundamental gradient estimate, which is now widely called the Li-Yau estimate, for any positive solution u(x, t) of the heat equation on a Riemannian manifold M(n = dimRM) and showed how the classical Harnack inequality can be derived from their gradient estimate. When M is complete and of nonnegative Ricci curvature, the Li-Yau estimate is sharp. Later in [H2], Richard Hamilton extended the Li-Yau estimate to the full matrix version of the Hessian estimate of u under the stronger assumptions that M is Ricci parallel and of nonnegative sectional curvature. In this paper, we consider positive solutions to the heat equation ( ∂

In [Y], Yau proposed to study the uniformization of complete Kahler manifolds with nonnegative curvature. In particular, one wishes to determine whether or not a complete Kahler manifold M with positive bisectional curvature is biholomorphic to Cm. See also [GW], [Si]. For this sake, it was further asked in [Y] whether or not the ring of the holomorphic functions with polynomial growth, which we denote by Op(M), is finitely generated, and whether or not the dimension of the spaces of holomorphic functions of polynomial growth is bounded from above by the dimension of the corresponding spaces of polynomials on Cm. This paper addresses the latter questions. We denote by 0?(M) the space of holomorphic functions of polynomial growth with degree d. (See Section 3 for the precise definition.) Then Op(M) = \Jd>0 Od(M). In this paper, we show that

We add two sections to (8) and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of (8), which reveals the relation between the entropy formula, (1.4) of (8), and the well-known Li-Yau's gradient estimate. As a by-product we obtain the sharp estimates on 'Nash's entropy' for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau's gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to R n .I n the second section we derive a dual entropy formula which, to some degree, connects Hamilton's entropy with Perelman's entropy in the case of Riemann surfaces. 1. The relation with Li-Yau's gradient estimates In this section we provide another derivation of Theorem 1.1 of (8) and discuss its relation with Li-Yau's gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash's 'entropy quantity' − M H log Hd vin the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of (9). Let u(x, t) be a positive solution to ∂ ∂t − � u(x, t) = 0 with M ud v= 1. We define

In this paper, we extend the general maximum principle in (NT3) to the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we exhibit complete Riemannian manifolds with bounded nonnegative sectional cur- vature of dimension greater than three such that the Ricci flow does not preserve the nonnegativity of the sectional curvature, even though the nonnegativity of the sectional curvature was proved to be preserved by Hamilton in dimension three. This fact is proved through a general splitting theorem on the complete family of metrics with nonnegative sectional curvature, deformed by the Ricci flow.

has been studied extensively and important applications were given. In [N1] and [N-S-T], the Poincare-Lelong equation has been solved under more general conditions than in [M-S-Y]. The conditions in [N-S-T] are more in line with the conditions in [Sh2-4]. Since a solution of (0.1) is a potential for the Ricci tensor, it is interesting to see if one can apply (0.1) to study solutions of (0.2). In this work, on the one hand we shall study the Kahler-Ricci flows by using solutions of the Poincare-Lelong equation. On the other hand, we will also refine some of the results in [Sh3, C-Z, C-T-Z] and give new applications. The hinge between the equations (0.1) and (0.2) is that by solving (0.1) one can then construct a function u(x, t) which satisfies the time-dependent heat equation ( ∂ ∂t − ∆)u(x, t) = 0 and the time-dependent Poincare-Lelong equation √−1∂∂u = Ricg(t) simultaneously. It then can simplify the study of (0.2) quite a bit. It also suggests some of the refined

In this paper, a sharp linear trace Li-Yau-Hamilton inequality for Kahler-Ricci flow is proved. The new inequality extends the previous trace Harnack inequality obtained by H.-D. Cao. We also establish sharp gradient estimates for the positive solution of the time-dependent heat equation for some cases. Finally, we apply this new linear trace Li-Yau-Hamilton inequality to study the Liouville properties of the plurisubharmonic functions on complete Kahler manifolds with bounded nonnegative holomorphic bisectional curvature.

In this article we study the metric property and the function theory of asymptotically locally Euclidean (ALE) Kähler manifolds. In particular, we prove the Ricci flatness under the assumption that the Ricci curvature of such manifolds is either nonnegative or nonpositive. The result provides a generalization of previous gap type theorems established by Greene and Wu, Mok, Siu and Yau, etc. It can also be thought of as a general positive mass type result. The method also proves the Liouville properties of plurisubharmonic functions on such manifolds. We also give a characterization of Ricci flatness of an ALE Kähler manifold with nonnegative Ricci curvature in terms of the structure of its cone at infinity.

This is an expository article based on the author’s lecture delivered at the conference Lie Theory and Its Applications in March 2011, UCSD. We discuss various notions of positivity and their relations with the study of the Ricci flow, including a proof of the assertion, due to Wolfson and the author, that the Ricci flow preserves the positivity of the complex sectional curvature. We discuss the examples of Wallach of the manifolds with positive pinched sectional curvature and the behavior of Ricci flow on some examples. Finally we discuss the recent joint work with Wilking on the manifolds with pinched flag curvature and some open problems.

Viscosity solutions are suitable notions in the study of nonlinear PDEs justified by estimates established via the maximum principle or the comparison principle. Here we prove that the isoperimetric profile functions of Riemannian manifolds with Ricci lower bound are viscosity super-solutions of some nonlinear differential equations. From these one can derive the isoperimetric inequalities of L\'evy-Gromov and B\'erard-Besson-Gallot, as well as some new comparison results.

We show that any minimal volume preserving map from the Euclidean plane into itself is a linear diffeomorphism. We derive this from a similar result on minimal diffeomorphisms. We also show that the classical Bernstein theorem on minimal graphs is a corollary of our result.

Here we shall use the same notations of the original paper. The references to equations, lemmas, propositions, theorems are with respect to the paper unless stated otherwise. The argument provided for the proof of Proposition 3.1 is not complete. It overlooks the fact that the solution 2-form η(x, t) of the problem (3.1) may contain non-trivial (2,0) and (0,2) parts. Hence Proposition 3.1 needs a correction. We thank Gilles Carron for pointing out this oversight in the original proof of Proposition 3.1 and his comments. We also thank Luen-Fai Tam for helpful related discussions. We refer interested reader to [5] for an alternate proof of Theorem 1.2. Since the preservation of the positivity for solution 2-forms solving the problem (3.1) in the paper, without knowing that the 2-form being (1,1)-type, seems elusive (even for the (1,1) part projection of the 2-form solutions) we replace (3.1) with the following initial boundary value problem on (1,1)form η. Let Ω be a smooth bounded domain in M . For a given point p ∈ ∂Ω , let ν be the unit outward normal in a small neighborhood near p. We

In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermitian-Einstein equations on holomorphic vector bundles over strictly pseudoconvex domains.

In this paper we study a nonlinear elliptic system of equations imposed on a map from a complete Hermitian (non-Kahler) manifold to a Riemannian manifold. This system is more appropriate to Hermitian geometry than the harmonic map system since it is compatible with the holomorphic structure of the domain manifold in the sense that holomorphic maps are Hermitian harmonic maps. It was first studied by Jost and Yau in [J-Y], and was applied to study the rigidity of compact Hermitian manifolds. We extend their existence and uniqueness results to the case where both domain and target manifolds are complete. Hopefully the results will be useful to study corresponding rigidity of complete Hermitian manifolds. Let M be a complex manifold with Hermitian metric (hαβ), and let N be a Riemannian manifold with metric (gij) and Christoffel symbols Γ i jk. A Hermitian harmonic map u : M → N satisfies the following elliptic system

In this paper, we develop a method of solving the Poincaré–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.

We prove a comparison theorem on the modulus of continuity of the solution of a heat equation with a drifting term on Bakry-Emery manifolds. A direct consequence of the result is an alternate proof of an eigenvalue comparison result of Bakry-Qian. Examples are given to show that the estimate is sharp. Discussions on an explicit lower estimate for the corresponding ODE and an application to the diameter lower bound for gradient shrinking solitons are also included.

By solving the Cauchy problem for the Hodge-Laplace heat equation for $d$-closed, positive $(1, 1)$-forms, we prove an optimal gap theorem for K��hler manifolds with nonnegative bisectional curvature which asserts that the manifold is flat if the average of the scalar curvature over balls of radius $r$ centered at any fixed point $o$ is a function of $o(r^{-2})$. Furthermore via a relative monotonicity estimate we obtain a stronger statement, namely a `positive mass' type result, asserting that if $(M, g)$ is not flat, then $\liminf_{r\to \infty} \frac{r^2}{V_o(r)}\int_{B_o(r)}\mathcal{S}(y)\, d��(y)>0$ for any $o\in M$.

In this article, by extending the method of Andrews and Clutterbuck (2011) [2] we prove a sharp estimate on the expansion modulus of the gradient of the logarithm of the parabolic kernel to the Schrodinger operator with convex potential on a bounded convex domain. The result improves an earlier work of Brascamp–Lieb which asserts the log-concavity of the parabolic kernel. We also give an alternate proof to a corresponding estimate on the first eigenfunction of the Schrodinger operator, obtained firstly by Andrews and Clutterbuck via the study of the asymptotics to a parabolic problem. Our proof is more direct via an elliptic maximum principle. An alternate proof of the fundamental gap theorem of Andrews and Clutterbuck (2011) [2] , by considering the quotient of moduli of continuity, is also obtained. Moreover we derive a Neumann eigenvalue comparison result and some other lower estimates on the first Neumann eigenvalue for Laplace operator with a drifting term, including an explicit estimate on a conjecture of P. Li.