1,161 research outputs found
Ricci flows with unbounded curvature
We show that any noncompact Riemann surface admits a complete Ricci flow
g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).Comment: 12 pages, 1 figure; updated reference
Ricci flows with bursts of unbounded curvature
Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metric, this Ricci flow always exists for all time and converges (after scaling by a factor 1/2t ) to this hyperbolic metric [GT11], i.e. our Ricci flow geometrises the surface. In this paper we show that there exist complete, bounded curvature initial metrics, including those conformal to a hyperbolic metric, which have subsequent Ricci flows developing unbounded curvature at certain intermediate times. In particular, when coupled with the uniqueness from [Top13], we find that any complete Ricci flow starting with such initial metrics must develop unbounded curvature over some intermediate time interval, but that nevertheless, the curvature must later become bounded and the flow must achieve geometrisation as t → ∞, even though there are other conformal deformations to hyperbolic metrics that do not involve unbounded curvature.
Another consequence of our constructions is that while our Ricci flow from [GT11] must agree initially with the classical flow of Hamilton and Shi in the special case that the initial surface is complete and of bounded curvature, by uniqueness, it is now clear that our flow lasts for a longer time interval in general, with Shi’s flow stopping when the curvature blows up, but our flow continuing strictly beyond in these situations.
All our constructions of unbounded curvature developing and then disappearing are in two dimensions. Generalisations to higher dimensions are then immediate
Flowing maps to minimal surfaces: Existence and uniqueness of solutions
We 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
Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schroedinger maps on R^2
We consider the Landau-Lifshitz equations of ferromagnetism (including the
harmonic map heat-flow and Schroedinger flow as special cases) for degree m
equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal
energy solutions converge to a harmonic map as t goes to infinity (asymptotic
stability), extending previous work down to degree m = 3. Due to slow spatial
decay of the harmonic map components, a new approach is needed for m=3,
involving (among other tools) a "normal form" for the parameter dynamics, and
the 2D radial double-endpoint Strichartz estimate for Schroedinger operators
with sufficiently repulsive potentials (which may be of some independent
interest). When m=2 this asymptotic stability may fail: in the case of
heat-flow with a further symmetry restriction, we show that more exotic
asymptotics are possible, including infinite-time concentration (blow-up), and
even "eternal oscillation".Comment: 34 page
Existence of Ricci flows of incomplete surfaces
We prove a general existence result for instantaneously complete Ricci flows
starting at an arbitrary Riemannian surface which may be incomplete and may
have unbounded curvature. We give an explicit formula for the maximal existence
time, and describe the asymptotic behaviour in most cases.Comment: 20 pages; updated to reflect galley proof correction
Harmonic maps from degenerating Riemann surfaces
We study harmonic maps from degenerating Riemann surfaces with uniformly
bounded energy and show the so-called generalized energy identity. We find
conditions that are both necessary and sufficient for the compactness in
and modulo bubbles of sequences of such maps.Comment: 27 page
Ricci flow for homogeneous compact models of the universe
Using quaternions, we give a concise derivation of the Ricci tensor for
homogeneous spaces with topology of the 3-dimensional sphere. We derive
explicit and numerical solutions for the Ricci flow PDE and discuss their
properties. In the collapse (or expansion) of these models, the interplay of
the various components of the Ricci tensor are studied. We dedicate this paper
to honor the work of Josh Goldberg.Comment: 18 pages, 2 figure
Up and down the number line: modelling collaboration in contrasting school and home environments
This paper is concerned with user modelling issues such as adaptive educational environments, adaptive information retrieval, and support for collaboration. The HomeWork project is examining the use of learner modelling strategies within both school and home environments for young children aged 5 – 7 years. The learning experience within the home context can vary considerably from school especially for very young learners, and this project focuses on the use of modelling which can take into account the informality and potentially contrasting learning styles experienced within the home and school
The effects of dietary fish oil on hepatic high density and low density lipoprotein receptor activities in the rat
AbstractRats were fed either a standard ration diet or that diet supplemented with 8% by wt of a marine fish oil or safflower oil. After 10 days, plasma triacylglycerols, total cholesterol, high density lipoprotein (HDL) cholesterol, hepatic cholesterol and fatty acid synthesis and hepatic low density lipoprotein (LDL) receptor activity were significantly depressed while HDL receptor activity was significantly increased in rats fed fish oil. Fish oil-induced effects on cholesterol metabolism in the rat therefore include reciprocal changes in the activities of hepatic LDL and HDL receptors
Interactions of asbestos-activated macrophages with an experimental fibrosarcoma
Supernatants from in vivo asbestos-activated macrophages failed to show any cytostatic activity against a syngeneic fibrosarcoma cell line in vitro. UICC chrysotile-induced peritoneal exudate cells also failed to demonstrate any growth inhibitory effect on the same cells in Winn assays of tumor growth. Mixing UICC crocidolite with inoculated tumor cells resulted in a dose-dependent inhibition of tumor growth; this could, however, be explained by a direct cytostatic effect on the tumor cells of high doses of crocidolite, which was observed in vitro
- …