Let M be an n -dimensional compact Willmore Lagrangian submanifold in a complex projective space CPn and let S and H be the squared norm of the second fundamental form and the mean curvature of M . Denote by ρ2 = S−nH2 the non-negative function on M , K and Q the functions which assign to each point of M the infimum of the sectional curvature and Ricci curvature at the point. We prove some inte...

‎we study curvature properties of four-dimensional lorentzian manifolds with two-symmetry property‎. ‎we then consider einstein-like metrics‎, ‎ricci solitons and homogeneity over these spaces‎‎.

Abstract. In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with Ricci(M) ≥ −k, k ∈ R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type L...

We consider Ricci flow invariant cones C in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to Ricci flow has its curvature operator which satsisfies R+ε I ∈ C at the initial time, then it satisfies R+Kε I ∈ C on some time interval depen...

For k 2, we exhibit complete k-curvature homogeneous neutral signature pseudoRiemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). All the local scalar Weyl invariants of these manifolds vanish. These manifolds are Ricci flat, Osserman, and Ivanov–Petrova. Mathematics Subject Classification (2000): 53B20.

One of the most important properties of a geometric flow is whether it preserves the positivity of various notions of curvature. In the case of the Kähler-Ricci flow, the positivity of the curvature operator (Hamilton [7]), the positivity of the biholomorphic sectional curvature (Bando [1], Mok[8]), and the positivity of the scalar curvature (Hamilton [4]) are all preserved. However, whether th...

In the study of conformai geometry, the method of elliptic partial differential equations is playing an increasingly significant role. Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition, see section 2) generalizing the conformai Laplacian, and their associated conformai invariants have been introduced. The conformally covariant powers of the La...

In the study of conformal geometry, the method of elliptic partial differential equations is playing an increasingly significant role. Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition, see section 2) generalizing the conformal Laplacian, and their associated conformal invariants have been introduced. The conformally covariant powers of the La...

In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in Bk × (0, T ) to the minimal fundamental solution of the conjugate heat equation as k → ∞. We will prove the uniqueness of the f...

Applying a well known result for attracting fixed points of biholomorphisms [4, 6], we observe that one immediately obtains the following result: if (Mn, g) is a complete non-compact gradient Kähler-Ricci soliton which is either steady with positive Ricci curvature so that the scalar curvature attains its maximum at some point, or expanding with non-negative Ricci curvature, then M is biholomor...