Willmore type inequality using monotonicity formulas

Monotonicity Formulas under Rescaled Ricci Flow

In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem 2.1. 1. Functionals Wek from rescaled Ricci flow point of view This is the research notes when the author wrote [Li07]. In the first section, we discuss the relation between functionals Wek(g, f, τ) and rescaled Ricci flow. In Theorem 4.2 [Li07] , we have defined funct...

The Brunn-Minkowski-Type Inequality

a cauchy-schwarz type inequality for fuzzy integrals

نامساوی کوشی-شوارتز در حالت کلاسیک در فضای اندازه فازی برقرار نمی باشد اما با اعمال شرط هایی در مسئله مانند یکنوا بودن توابع و قرار گرفتن در بازه صفر ویک می توان دو نوع نامساوی کوشی-شوارتز را در فضای اندازه فازی اثبات نمود.

Local Monotonicity and Mean Value Formulas for Evolving Riemannian Manifolds

We derive identities for general ‡ows of Riemannian metrics that may be regarded as local mean-value, monotonicity, or Lyapunov formulae. These generalize previous work of the …rst author for mean curvature ‡ow and other nonlinear di¤usions. Our results apply in particular to Ricci ‡ow, where they yield a local monotone quantity directly analogous to Perelman’s reduced volume ~ V and a local id...

Heat-flow Monotonicity Related to the Hausdorff–young Inequality

It is known that if q is an even integer then the L(R) norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres “simultaneously slide” to the origin. We provide explicit examples to show that this monotonicity property fails dramatically if q > 2 is not an even integer. These results are equivalent, upon rescaling, to similar sta...