The Conical K(a|")hler-Ricci Flow On Fano Manifolds

Kahler-Ricci flow, first introduced by H.D. Cao[9] in the1980s, is an important subject in geometry analysis and partial differential equation. After being researched and improved by H.D. Cao, A. Chau, X.X. Chen, G. Perelman, D. Phong, W.X. Shi, J. Song, J. Sturm, G. Tian, B. Weinkove and X.H. Zhu etc. in the last several years, the study of the Kahler-Ricci flow has developed into a vast field in its own right. There have been several different avenues of research involving this flow, including:the ex-istence of Kahler-Einstein metrics on manifolds; the classification of Kahler manifolds and extensions of the flow to non-Kahler settings. Kahler-Ricci flow plays an important role in researching and solving these problems.In this paper, we study a kind of singular Kahler-Ricci flows-conical Kahler-Ricci flow. We mainly discuss its long-time existence and convergence. For these problems, the smooth Kahler-Ricci flow is more developed than the conical one. In our research, we combine the conical Kahler-Ricci flow with the twisted Kahler-Ricci flow. First of all, we obtain a long-time solution to the conical Kahler-Ricci flow by limiting a sequence of twisted Kahler-Ricci flows. After an appropriate improvement, we prove the uniform Perelman’s estimates along a sequence of twisted Kahler-Ricci flows. At last, we discuss the convergence of the conical Kahler-Ricci flow on Fano manifold. We obtain the following main theorem:定理0.1.Let M be a Fano manifold with complex dimension n, ω0be a smooth Kahler metric in2πc1(M), h be a smooth Hermitian metric on the anti-canonical bundle-Km with curvature ω0, D∈|-Km|be a smooth divisor on M and s be the defining section of D. For any β∈(0,1), the conical Kahler-Ricci flow admits a unique long-time solution ω(t).Furthermore, if there exists a conical Kdhler-Einstein metric ωβ,D with cone angle 2π/3along divisor D, then the long-time solution ui(t) must converge to the metric in Cloc∞topology outside the divisor D and globally in the sense of currents on M.The proof of Theorem0.1depends on some important results, such as the relevant regularity theory and the uniform Perelman’s estimates etc. In this paper, we prove these results. Now, we list them all here.In general, the global regularity estimates of the solution to equation are very im-portant. If we can not obtain the satisfying global regularity estimates, then considering the local higher order estimates is also significant. According to the actual need in the proof of the long-time existence of the conical Kahler-Ricci flow, we study the local higher order estimates of the twisted Kahler-Ricci flows. Notably, in order to get the uniform C∞estimates of the solutions Φε(t) to a sequence of Monge-Ampere equa-tions on K x [0, T](?)(?)C (M\D)×[0, T](here the estimates including the derivative estimates of space variables and time variable; the uniformity means that the estimates independent of ε and t), after combining the derivative estimates of curvature,(?) and Φε(t), we argue this problem based on the elliptic Schauder estimate which is su-perior to the parabolic one, because the latter can only provide us with a local uniform C∞estimates on Br×[δ, T] for some δ>0, and the fact that these estimates depend on δ.Let where▽denotes the covariant derivative with respect to the smooth Kahler metric ω0, gφ be the metric tensor corresponds to the metric for-m ωφ, dVω0=ω0n)/(n!) be the volume element with respect to the metric ω0and satisfy1/V∫MdVω0=1,D be the covariant derivative with respect to the metric ωφ. We also denote the curvature tensor of ωφ by Rmφ.命题0.2.Let φ(·,t) be a solution of the parabolic Monge-Ampere equation where f is the twisted Ricci potential with respect to the metric ω0, i.e. f satisfies satisfies for some constant N. Then there exist constant C′and C″such that on Br/2(p)×[0,T]. The constant C′depends only on ω0, N,γ,||φ(·,0)||C3(Br,(p,)) and||θ||C1(Br,(p)); constant C″depends only onwo, N, γ,||φ(·,0)||C4(Br(p)) and||θ||C2(Br.(p)).Furthermore, there exist constants Ck1, Ck2and Ck3such that for any k≥0on Br/2(p)×[0,T]. Here constants Ck1, Ck2and Ck3depend only on ω0, N, γ,||φ(·,0)||Ck+4(Br,(p)),||θ||Ck+2(Br,(p)),||φ||C0(Br(p)×[0,T])and||f||C0(Br(p)).As the application of Proposition0.2, we consider the long-time existence of the general conical Kahler-Ricci flow In equation (13), we assume that divisor D∈|-λKM|(λ∈Q), μ=1-(1-β)λ, and h is a smooth Hennitian metric on the line bundle-λKM with curvature,λω0.Let Fo be the Ricci potential with respect to the metric ω0, i.e. F0satisfies and By using the long-time existence of the twisted Kahler-Ricci flow and Proposition0.2, we have the following theorem: 定理0.3.Assume β∈(0,1). Then the solution Φε of the equation converge to a solution ip of the equation in the Cloc∞topology on (M\D)×[0,+∞). Meanwhile, is the unique long-time solution to the conical Kahler-Ricci flow (13).After that, combining the twisted Kahler-Ricci flow, we study the convergence of the conical Kahler-Ricci flow (13) when the twisted Chern class Ci,β(M)=2πc1(M)-(1-β)[D] is positive (i.e. μ>0). To reader’s convenience, we just discuss this prob-lem when λ=1, i.e.μ=β."We first prove the uniform Perelman’s estimates along the twisted Kahler-Ricci flow where is a smooth closed positive (1,1)-form.By further study on the initial condition on which the Perelman’s estimates mainly depend in the twisted Kahler-Ricci flow case, we know that these estimates mainly depend on the bound of the initial twisted scalar curvature R(ge(0))-trgε (0)θε and Sobolev constant Cs(M,gε(0)). But the initial twisted scalar curvature may not be bounded uniformly when β∈(1/2,1). In order to overcome this difficulty, we consider the uniform boundness of the relevant geometric quantities at time t=1instead of t=0. Through this improvement, for any β∈(0,1), we obtain the uniform Perelman’s estimates along the twisted Kahler-Ricci flow (17) when t≥1. This is very important when we consider the convergence of the conical Kahler-Ricci flow (11). 定理0.4.Let gε(t) be a solution of the twisted Kahler Ricci flow, i.e. the corresponding form ωε(t) satisfies the equation (17), uε(t)∈C∞(M) be the twisted Ricci potential satisfying equation and normalization1/v∫Me-uε(t)dVεt=1. Then for any β∈(0,1), there exists a uniform constant C, such that hold for any t≥1and ε>0, where R(gε(t))-trgε(t)θε and diam(M, gε(t)) are the twisted scalar curvature and diameter of the manifold respectively with respect to the metric gε(t).Based on the above uniform estimates, we can give a well initial value to the equation (15) when μ=β. After that, we can obtain the uniform C0estimates of Φε(t) by the properness of Mabuchi κ-energy functional Mωo, θε on the smooth Kahler potential space H(ω0), the uniform Sobolev inequality and the uniform weight-ed Poincare inequality along the twisted Kahler-Ricci flow (17). It is well known that the C0estimate is important in the arguments of the convergence of the smooth Kahler-Ricci flow. Likewise, the above uniform C0estimates also play a key role in the convergence of the conical Kahler-Ricci flow (11).定理0.5.Let Φε(t) be a solution to the equation (15) with initial value (19) when μ=β. If the twisted Mabuchi κ-energy functional Mω0，θε is uniformly proper on the space H(ω0), then there exists a uniform constant C, such that for any t≥0and ε>0. Now we consider the convergence of the conical Kahler-Ricci flow. If there ex-ists a conical Kahler-Einstein metric ωβ,D with cone angle2πβ along divisor D. By using the relationship between the existence of conical Kahler-Einstein metric and the properness of some energy functional, Theorem0.5and the locally uniform regularity estimates (see Theorem0.2), we can get a time sequence{ti} by the diagonal rule such that ω(ti) converge to a conical Kahler-Einstein metric ω∞with cone angle27πβ along divisor D. Then by the uniqueness of the conical Kahler-Einstein metric, we prove that ω(t) must converge to ω∞=ωβ,D in Cloc∞topology outside divisor D and globally in the sense of currents. In fact, we obtain the following convergence theorem of the conical Kahler-Ricci flow (11):定理0.6.Let β∈(0,1). Assume that there exists a conical Kahler-Einstein meric ωβ,D with cone angle2πβ along divisor D, then the flow (11) converges to the conical Kahler-Einstein meric ωβ,D in Cloc∞topology outside D and globally in the sense of currents.