A uniform logarithmic Sobolev inequality,a uniform Sobolev inequality and a uniformκ-noncollapsing estimate along the Ricci flow are established in the situation that a certain smallest eigenvalue for the initial met...A uniform logarithmic Sobolev inequality,a uniform Sobolev inequality and a uniformκ-noncollapsing estimate along the Ricci flow are established in the situation that a certain smallest eigenvalue for the initial metric is zero.展开更多
In this note we present various extensions of Obata’s rigidity theorem concerning the Hessian of a function on a Riemannian manifold.They include general rigidity theorems for the generalized Obata equation,and hyper...In this note we present various extensions of Obata’s rigidity theorem concerning the Hessian of a function on a Riemannian manifold.They include general rigidity theorems for the generalized Obata equation,and hyperbolic and Euclidean analogs of Obata’s theorem.Besides analyzing the full rigidity case,we also characterize the geometry and topology of the underlying manifold in more general situations.展开更多
Based on Perelman’s entropy monotonicity,uniform logarithmic Sobolev inequalities along the Ricci flow are derived.Then uniform Sobolev inequalities along theRicci floware derived via harmonic analysis of the integra...Based on Perelman’s entropy monotonicity,uniform logarithmic Sobolev inequalities along the Ricci flow are derived.Then uniform Sobolev inequalities along theRicci floware derived via harmonic analysis of the integral transform of the relevant heat operator.These inequalities are fundamental analytic properties of the Ricci flow.They are also extended to the volume-normalized Ricci flow and the Kähler-Ricci flow.展开更多
A number of results about deriving further Sobolev inequalities from a given Sobolev inequality are presented.Various techniques are employed,including Bessel potentials and Riesz transforms.Combining these results wi...A number of results about deriving further Sobolev inequalities from a given Sobolev inequality are presented.Various techniques are employed,including Bessel potentials and Riesz transforms.Combining these results with theW1,2 Sobolev inequality along the Ricci flow established by the author in earlier papers then yields various new Sobolev inequalities along the Ricci flow.展开更多
文摘A uniform logarithmic Sobolev inequality,a uniform Sobolev inequality and a uniformκ-noncollapsing estimate along the Ricci flow are established in the situation that a certain smallest eigenvalue for the initial metric is zero.
文摘In this note we present various extensions of Obata’s rigidity theorem concerning the Hessian of a function on a Riemannian manifold.They include general rigidity theorems for the generalized Obata equation,and hyperbolic and Euclidean analogs of Obata’s theorem.Besides analyzing the full rigidity case,we also characterize the geometry and topology of the underlying manifold in more general situations.
文摘Based on Perelman’s entropy monotonicity,uniform logarithmic Sobolev inequalities along the Ricci flow are derived.Then uniform Sobolev inequalities along theRicci floware derived via harmonic analysis of the integral transform of the relevant heat operator.These inequalities are fundamental analytic properties of the Ricci flow.They are also extended to the volume-normalized Ricci flow and the Kähler-Ricci flow.
文摘A number of results about deriving further Sobolev inequalities from a given Sobolev inequality are presented.Various techniques are employed,including Bessel potentials and Riesz transforms.Combining these results with theW1,2 Sobolev inequality along the Ricci flow established by the author in earlier papers then yields various new Sobolev inequalities along the Ricci flow.
