Category Archives: Curvature dimension inequalities

Lecture 25. The Sobolev inequality proof of the Myer’s diameter theorem

It is a well-known result that if is a complete -dimensional Riemannian manifold with , for some , then has to be compact with diameter less than . The proof of this fact can be found in any graduate book … Continue reading

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Lecture 24. Sharp Sobolev inequalities

In this Lecture, we are interested in sharp Sobolev inequalities in positive curvature. Let be a complete and -dimensional Riemannian manifold such that where . We assume . As we already know from Lecture 15 , we have , but … Continue reading

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Lecture 23. The isoperimetric inequality

In this Lecture, we study in further details the connection between volume growth of metric balls, heat kernel upper bounds and the Sobolev inequality. As we shall see, on a manifold with non negative Ricci curvature, all these properties are … Continue reading

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Lecture 22. Sobolev inequality and volume growth

In this Lecture, we show how Sobolev inequalities on a Riemannian manifold are related to the volume growth of metric balls. The link between the Hardy-Littlewood-Sobolev theory and heat kernel upper bounds is due to Varopoulos, but the proof I … Continue reading

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Lecture 21. The PoincarĂ© inequality on domains

Let be a complete Riemannian manifold and be a non empty bounded set. Let be the set of smooth functions such that for every , It is easy to see that is essentially self-adjoint on . Its Friedrichs extension, still … Continue reading

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Lecture 20. Upper and lower heat kernel Gaussian bounds

In this short Lecture, as in the previous one, we consider a complete and -dimensional Riemannian manifold with non negative Ricci curvature. The volume doubling property that was proved is closely related to sharp lower and upper Gaussian bounds that … Continue reading

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Lecture 19. Volume doubling property

In this Lecture we consider a complete and -dimensional Riemannian manifold with non negative Ricci curvature. Our goal is to prove the following fundamental result, which is known as the volume doubling property. Theorem: There exists a constant such that … Continue reading

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