303 research outputs found
L-optimal transportation for Ricci flow
No full textWe introduce the notion of L-optimal transportation, and use it to construct a natural monotonic quantity for Ricci flow which includes a selection of other monotonicity results, including some key discoveries of Perelman [13] (both related to entropy and to L-length) and a recent result of McCann and the author [11]
The canonical shrinking soliton associated to a Ricci flow
No full textTo every Ricci flow on a manifold over a time interval , we associate a shrinking Ricci soliton on the space-time . We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author Topping (J Reine Angew Math 636:93-122, 2009), and McCann and the second author (Am J Math 132:711-730, 2010); we briefly survey the link between these subjects
The canonical shrinking soliton associated to a Ricci flow
No full textTo every Ricci flow on a manifold over a time interval IR−, we associate a shrinking Ricci soliton on the space–time I. We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author Topping (J Reine Angew Math 636:93–122, 2009), and McCann and the second author (Am J Math 132:711–730, 2010); we briefly survey the link between these subjects
Flowing maps to minimal surfaces: Existence and uniqueness of solutions
No full textWe study the new geometric flow that was introduced in [11] that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (weak) solution exists for as long as the metrics remain in a bounded region of moduli space, i.e. as long as the flow does not collapse a closed geodesic in the domain manifold to a point. Furthermore, we prove that this solution is unique in the class of all weak solutions with non-increasing energy. This work complements the paper [11] of Topping and the author where the flow was introduced and its asymptotic convergence to branched minimal immersions is discussed
The harmonic map heat flow from surfaces
No full textWe present a study of the harmonic map heat flow of Eells and Sampson in the case that
the domain manifold is a surface. Particular emphasis has been placed on the singularities
which may occur, as described by Struwe, and the analysis of the flow despite these.
In Chapter 1 we give a brief introduction to the theory of harmonic maps and their flow.
Further details are to be found in [9] and [10]. In the case that the domain manifold is a
surface we describe the existence theory for the heat flow and the theory of bubbling.
In Chapter 2 we investigate the question of the uniformity in time of the convergence of
the heat flow to a bubble tree at infinite time. In Section (2.1) (page 28) we give the first
example of a non-uniform flow. In contrast, Theorem (2.2) (page 30) provides conditions
under which the convergence is uniform and any bubbles which form are rigid.
In Chapter 3 we give the first example of a nontrivial bubble tree - in other words we give
a flow in which more than one bubble develops at the same point at infinite time.
In Chapter 4 we discuss in what sense two flows are close when their initial maps are
close. We formulate this question in various ways, providing examples of instability and
an `infinite time' stability result (Theorem (4.2), page 56) using techniques developed in
Chapter 2.
From the theory of bubbling as described in Chapter 1, if an initial map has less energy
than is required for a bubble, then the subsequent flow cannot blow up. In Chapter 5
we ask conversely whether given enough energy for a bubble, we can find an initial map
leading to blow-up.
In the appendix we outline a plausible construction of a flow which can be analysed at
two different sequences of times to give convergence to two different bubble trees, with
different numbers of bubbles
Asymptotics of the Teichmüller harmonic map flow
No full textThe Teichmüller harmonic map flow, introduced by Rupflin and Topping (2012) [11], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case that the domain metric of the flow degenerates at infinite time. We obtain a branched minimal immersion from the degenerate domain
In Claudel's footsteps
No full textBook synopsis: Strange and exotic, seductive and threatening, the Orient has always been an enchanted space for the West. But this is a space, theorists argue, that has been ‘Orientalized’ by the West, constructed upon a system of knowledge and power which defines the West as much as this ‘Other’. Within Western cultures, the French encounter with the Orient has been extraordinarily rich and varied, from the experiences of the first pilgrims to the challenges posed for the identity of modern-day France by its ethnic minorities. This collection of interdisciplinary essays explores the range of French and francophone encounters with the East from the medieval period to the present day. The contributions encompass a variety of Orients, both geographical and generic: the Orients of the visual arts, of historicist discourse, of fiction and travel writing. They consider not only those artists we immediately associate with the East, such as Nerval or Fromentin, but also those, like Proust, whose work appears firmly rooted in the West. They also provide new insights into the less familiar works of long-celebrated authors like Flaubert and more recently acclaimed writers such as Bouvier and Djebar
Biomechanical Effects of a Dynamic Topping off Instrumentation in a Long Rigid Pedicle Screw Construct
No full textStudy Design: Biomechanical ex vivo study. Objective: To determine if topping off instrumentation can reduce the hypermobility in the adjacent segments when compared with the classic rigid spinal instrumentation. Summary of the Background Data: Long rigid instrumentation might increase the mechanical load in the adjacent segments, the resulting hypermobility, and the risk for adjacent segment disease. Topping off instrumentation intends to reduce the hypermobility at the adjacent level by more evenly distributing segmental motion and, thereby, potentially mitigating adjacent level disease. Materials and Methods: Eight human spines (Th12-L5) were divided into 2 groups. In the rigid group, a 3-segment metal rod instrumentation (L2-L5) was performed. The hybrid group included a 2-segment metal rod instrumentation (L3-L5) with a dynamic topping off instrumentation (L2-L3). Each specimen was tested consecutively in 3 different configurations: native (N=8), 2-segment rod instrumentation (L3-L5, N=8), 3-segment instrumentation (rigid: N=4, hybrid: N=4). For each configuration the range of motion (ROM) of the whole spine and each level was measured by a motion capture system during 5 cycles of extension-flexion (angle controlled to ±5 degrees, 0.1 Hz frequency, no preload). Results: In comparison with the intact spine, both the rigid 3-segment instrumentation and the hybrid instrumentation significantly reduced the ROM in the instrumented segments (L2-L5) while increasing the movement in the adjacent segment L1-L2 (P=0.002, η 2 =0.82) and in Th12-L1 (P<0.001, η 2 =0.90). There were no ROM differences between the rigid and hybrid instrumentation in all segments. Conclusions: Introducing the dynamic topping off did not impart any significant difference in the segmental motion when compared with the rigid instrumentation. Therefore, the current biomechanical study could not show a benefit of using this specific topping off instrumentation to solve the problem of adjacent segment disease
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