Results 1  10
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18
Lectures on stability and constant scalar curvature
, 2008
"... An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kähler metrics of constant scalar curvature. Besides classical notions such as ChowMumford stability, the emphasis is on several new stability conditions, such as Kstability, D ..."
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Cited by 13 (2 self)
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An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kähler metrics of constant scalar curvature. Besides classical notions such as ChowMumford stability, the emphasis is on several new stability conditions, such as Kstability, Donaldson’s infinitedimensional GIT, and conditions on the closure of orbits of almostcomplex structures under the diffeomorphism group. Related analytic methods are also discussed, including estimates for energy functionals, TianYauZelditch approximations, estimates for moment maps, complex MongeAmpère equations and pluripotential theory, and the KählerRicci flow.
On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow
, 708
"... Abstract. In this note we construct Nadel multiplier ideal sheaves using the Ricci flow on Fano manifolds. This extends a result of Phong, ˇ Seˇsum and Sturm. These sheaves, like their counterparts constructed by Nadel for the continuity method, can be used to obtain an existence criterion for Kähl ..."
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Cited by 10 (1 self)
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Abstract. In this note we construct Nadel multiplier ideal sheaves using the Ricci flow on Fano manifolds. This extends a result of Phong, ˇ Seˇsum and Sturm. These sheaves, like their counterparts constructed by Nadel for the continuity method, can be used to obtain an existence criterion for KählerEinstein metrics. 1 Introduction. In this note we construct Nadel multiplier ideal sheaves on Fano manifolds that do not admit KählerEinstein metrics, using the Ricci flow. The result is a simple consequence of the uniformity of the Poincaré and Sobolev inequalities along the flow. This allows to obtain another proof of the convergence of the Ricci flow on a certain class of Fano manifolds.
The Log Entropy Functional Along The Ricci Flow
, 2007
"... 2. The log entrop functional 3. Monotonicity of the log entropy and the logarithmic Sobolev constant ..."
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Cited by 4 (0 self)
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2. The log entrop functional 3. Monotonicity of the log entropy and the logarithmic Sobolev constant
Compactness results for the KählerRicci flow
"... We consider the KählerRicci flow ∂ ∂tgi¯j = gi¯j − Ri¯j on a compact Kähler manifold M with c1(M)> 0, of complex dimension k. We prove the ǫregularity lemma for the KählerRicci flow, based on Moser’s iteration. Assume that the Ricci curvature and ∫ M Rmk dVt are uniformly bounded along the flow ..."
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Cited by 3 (0 self)
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We consider the KählerRicci flow ∂ ∂tgi¯j = gi¯j − Ri¯j on a compact Kähler manifold M with c1(M)> 0, of complex dimension k. We prove the ǫregularity lemma for the KählerRicci flow, based on Moser’s iteration. Assume that the Ricci curvature and ∫ M Rmk dVt are uniformly bounded along the flow. Using the ǫregularity lemma we derive the compactness result for the KählerRicci flow. Under our assumptions, if k ≥ 3 in addition, using the compactness result we show that Rm  ≤ C holds uniformly along the flow. This means the flow does not develop any singularities at infinity. We use some ideas of Tian from [22] to prove the smoothing property in that case. 1
UNIFORM SOBOLEV INEQUALITIES FOR MANIFOLDS EVOLVING BY RICCI FLOW
, 2007
"... Abstract. Let M be a compact ndimensional manifold, n ≥ 2, with metric g(t) evolving by the Ricci flow ∂gij/∂t = −2Rij in (0, T) for some T ∈ R + ∪ {∞} with g(0) = g0. Let λ0(g0) be the first eigenvalue of the operator −∆g0 recent result of R. Ye and prove uniform logarithmic Sobolev inequality an ..."
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Cited by 3 (0 self)
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Abstract. Let M be a compact ndimensional manifold, n ≥ 2, with metric g(t) evolving by the Ricci flow ∂gij/∂t = −2Rij in (0, T) for some T ∈ R + ∪ {∞} with g(0) = g0. Let λ0(g0) be the first eigenvalue of the operator −∆g0 recent result of R. Ye and prove uniform logarithmic Sobolev inequality and uniform Sobolev inequalities along the Ricci flow for any n ≥ 2 when either T < ∞ or λ0(g0)> 0. As a consequence we extend Perelman’s local κnoncollapsing result along the Ricci flow for any n ≥ 2 in terms of upper bound for the scalar curvature when either T < ∞ or λ0(g0)> 0. R(g0) 4 with respect to g0. We extend a Recently there is a lot of studies on Ricci flow on manifolds because it is an important tool in understanding the geometry of manifolds [H1–3], [Hs1–3], [KL], [MT], [P1], [P2]. On the other hand given any compact ndimensional manifold M, n ≥ 2, with a fixed metric g it is known that Sobolev inequalities hold [He]. More specifically for any q ∈ [1, n) and p satisfying 1 p
THE MODIFIED KÄHLERRICCI FLOW AND SOLITONS
, 2008
"... We investigate the KählerRicci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a KählerRicci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified ..."
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Cited by 2 (0 self)
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We investigate the KählerRicci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a KählerRicci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy.
Stability of KählerRicci flow
, 801
"... Abstract We prove the convergence of KählerRicci flow with some small initial curvature conditions. As applications, we discuss the convergence of KählerRicci flow when the complex structure varies on a KählerEinstein manifold. Keywords KählerRicci flow; KählerEinstein metrics, stability. 2000 ..."
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Abstract We prove the convergence of KählerRicci flow with some small initial curvature conditions. As applications, we discuss the convergence of KählerRicci flow when the complex structure varies on a KählerEinstein manifold. Keywords KählerRicci flow; KählerEinstein metrics, stability. 2000 Mathematics Subject Classification: 53C44, 32Q20.
M where r =
, 710
"... Abstract. Let (Mn, g0) (n odd) be a compact Riemannian manifold with positive scalar curvature. Assume the solution g(t) to the normalized Ricci flow with initial data (Mn, g0) satisfies R(g(t)) ≤ C and R M Rm(g(t))n/2dµt ≤ C uniformly on the maximal time interval of existence for a constant C. T ..."
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Abstract. Let (Mn, g0) (n odd) be a compact Riemannian manifold with positive scalar curvature. Assume the solution g(t) to the normalized Ricci flow with initial data (Mn, g0) satisfies R(g(t)) ≤ C and R M Rm(g(t))n/2dµt ≤ C uniformly on the maximal time interval of existence for a constant C. Then we show that the solution converges along a subsequence to a shrinking Ricci soliton. Since Hamilton’s seminal work [H] Ricci flow has been an important tool used extensively in geometry and topology. In particular, there is the recent breakthrough of Perelman [P1],[P2]. In this short note we prove a convergence result for odd dimensional volumenormalized
unknown title
, 710
"... Abstract. Let (Mn, g0) (n odd) be a compact Riemannian manifold with λ(g0)> 0, where λ(g0) is the lowest eigenvalue of the operator − ∆ + R(g0), 4 and R(g0) is the scalar curvature of (Mn, g0). Assume the solution g(t) to the normalized Ricci flow with initial data (Mn R, g0) satisfies R(g(t))  ≤ ..."
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Abstract. Let (Mn, g0) (n odd) be a compact Riemannian manifold with λ(g0)> 0, where λ(g0) is the lowest eigenvalue of the operator − ∆ + R(g0), 4 and R(g0) is the scalar curvature of (Mn, g0). Assume the solution g(t) to the normalized Ricci flow with initial data (Mn R, g0) satisfies R(g(t))  ≤ C and M Rm(g(t))n/2dµt ≤ C uniformly on the maximal time interval of existence of g(t) for a constant C. Then we show that the solution converges along a subsequence to a shrinking Ricci soliton. Since Hamilton’s seminal work [H] Ricci flow has been an important tool used extensively in geometry and topology. In particular, there is the recent breakthrough of Perelman [P1],[P2]. In this short note we prove a convergence result for odd dimensional volumenormalized