Results 1  10
of
10
The metric geometry of the manifold of Riemannian metrics
"... Abstract. We prove that the L 2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finitedimensional manifold induces a metric space structure. As the L 2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove sev ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We prove that the L 2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finitedimensional manifold induces a metric space structure. As the L 2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finitedimensional manifold or a strong Riemannian metric on a Hilbert manifold. 1.
RICCI FLOW AND THE METRIC COMPLETION OF THE SPACE OF KÄHLER METRICS
, 2012
"... We consider the space of Kähler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are equivalent. We also determine the metric completion of the space of K ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
We consider the space of Kähler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are equivalent. We also determine the metric completion of the space of Kähler metrics, making contact with recent generalizations of the Calabi–Yau Theorem due to Dinew and Guedj–Zeriahi. As an application, we obtain a new analytic stability criterion for the existence of a Kähler–Einstein metric on a Fano manifold in terms of the Ricci flow and the distance function. We also prove that the Kähler–Ricci flow converges as soon as it converges in the metric sense.
SOBOLEV METRICS ON THE MANIFOLD OF ALL RIEMANNIAN METRICS
"... Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as described first by [11]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellpo ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as described first by [11]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellposed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics. 1.
Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics. arXiv
, 1104
"... ar ..."
Sobolev metrics on the Riemannian manifold of all Riemannian metrics
, 2010
"... Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as decribed first by [10]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellpos ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as decribed first by [10]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellposed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of this metrics. 1.
Contents
, 2011
"... The HahnBanach Theorem is one of the most fundamental results in functional analysis. We present a fully formal proof of two versions of the theorem, one for general linear spaces and another for normed spaces. This development is based on simplytyped classical settheory, as provided by ..."
Abstract
 Add to MetaCart
The HahnBanach Theorem is one of the most fundamental results in functional analysis. We present a fully formal proof of two versions of the theorem, one for general linear spaces and another for normed spaces. This development is based on simplytyped classical settheory, as provided by
GEODESICS, DISTANCE, AND THE CAT(0) PROPERTY FOR THE MANIFOLD OF RIEMANNIAN METRICS
"... Abstract. Given a fixed closed manifoldM, we exhibit an explicit formula for the distance function of the canonical L2 Riemannian metric on the manifold of all smooth Riemannian metrics on M. Additionally, we examine the (metric) completion of the manifold of metrics with respect to the L2 metric an ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Given a fixed closed manifoldM, we exhibit an explicit formula for the distance function of the canonical L2 Riemannian metric on the manifold of all smooth Riemannian metrics on M. Additionally, we examine the (metric) completion of the manifold of metrics with respect to the L2 metric and show that there exists a unique minimal path between any two points. This path is also given explicitly. As an application of these formulas, we show that the metric completion of the manifold of metrics is a CAT(0) space. 1.
Contemporary Mathematics Optimal Riemannian
"... metric for a volumorphism and a mean ergodic theorem in complete global Alexandrov nonpositively curved spaces ..."
Abstract
 Add to MetaCart
metric for a volumorphism and a mean ergodic theorem in complete global Alexandrov nonpositively curved spaces