On short time existence for the planar network flow

We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature flow. We also show a pseudolocality theorem for mean curvature flow in any codimension, assuming only that the initial submanifold can be locally written as a graph with sufficiently small Lipschitz constant.Comment: Final version, to appear in Journal of Differential Geometry. 51 page

Entropy and reduced distance for Ricci expanders

Perelman has discovered two integral quantities, the shrinker entropy \cW and the (backward) reduced volume, that are monotone under the Ricci flow \pa g_{ij}/\pa t=-2R_{ij} and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The {\it expanding entropy} \ctW is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnack-like quantity for the conjugate heat equation, and leads to functionals $\mu_+$ and $\nu_+$. The {\it forward reduced volume} $\theta_+$ is monotone in general and constant exactly on expanders. A natural conjecture asserts that $g(t)/t$ converges as $t\to\infty$ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expanders; these include \Vol(g)/t^{n/2} (Hamilton) and $\bar\lambda$ (Perelman), as well as our new quantities. In general, we show that if \Vol(g) grows like $t^{n/2}$ (maximal volume growth) then \ctW, $\theta_+$ and $\bar\lambda$ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture

Ohutfilmihaiduttimen mallintaminen ionisen nesteen kierrättämiseksi

World demand for textiles is on the rise and there is a need for fiber source that does not require arable land. Ioncell-F is a novel method of producing fiber from pulp. It uses ionic liquid [DBNH][OAc] to dissolve cellulose. Ionic liquid is expensive so it has to be recycled to make the process economical. Ionic liquid is thermally unstable, therefore the recycling has to be done with moderate temperatures. One way to separate and recycle ionic liquid from water is thin film evaporation. The purpose of this study was to model the evaporation of water from water/[DBNH][OAc] mixture in an agitated thin film evaporator in flowsheet simulator Aspen Plus. Accurate modeling of the evaporator is needed to design and optimize the recycling process. The study also studied the applicability of multiple-effect evaporation. Various modeling approaches were studied to simulate the thin film evaporator and an Aspen Plus model was developed based on batch distillation theory. The performance of the model was compared to an earlier developed model based on flash drum model and experimental data. The batch model gave more accurate results than the often used flash model. The model did not include the hydrolysis product of the ionic liquid, implementation of which should be the focus in future works. A case study was conducted and the applicability of double-effect evaporation was tested with the model. A rapid boiling point elevation at low water concentrations made it harder to implement multiple-effect evaporation in recycling of the ionic liquid. It could be done with right pressure and temperature settings for most of the evaporation, with one additional evaporator to achieve desired water content. Multiple-effect evaporation proved useful in both achieving purer vapor and lowering the total required heating power.Maailmanlaajuinen tekstiilin kysyntä on kasvussa, ja tekstiilin lähteeksi tarvitaan raaka-aineita, jotka eivät vie tilaa viljeltävältä maalta. Ioncell-F on uusi prosessi, joka voi tuottaa kuitua sellusta. Se käyttää ionista nestettä [DBNH][OAc] liuottaakseen selluloosan. Ioninen neste on kallista, joten se täytyy erottaa vedestä ja kierrättää, jotta prosessi on kannattava. Se kuitenkin myös hajoaa korkeissa lämpötiloissa, joten erotus täytyy tehdä matalissa lämpötiloissa. Yksi keino tähän on ohutfilmihaihdutin. Tämän työn tarkoituksena oli mallintaa veden haihdutusta veden ja ionisen nesteen seoksesta ohutfilmihaihduttimessa Aspen-Plus ohjelmalla. Haihduttimen tarkka mallinnus on tärkeää kierrätysprosessin suunnittelemiseksi ja optimoinniksi. Työssä myös selvitettiin, olisiko monivaihelauhdutus mahdollista. Erilaisia malllinnusvaihtoehtoja käsiteltiin ja panostislaukseen pohjautuva malli valittiin keskittymispohjaksi. Aspen Plus malli kehitettiin panostislausteorian pohjalta, ja sen tuloksia verrattiin aiemmin kehitettyyn flash-malliin sekä kokeellisiin tuloksiin. Panostislausmallin tulokset olivat lähempänä kokeellisia, kuin flash-malli. Kummastakin mallista kuitenkin puuttuu ionisen nesteen hydrolyysituote, mikä tuo epätarkkuutta mallinnukseen. Sen sisältäminen mallinnukseen tulisi olla seuraavien tutkimusten kohde. Panostislausmallille tehtiin tapaustutkimus ja monivaihelauhdutuksen sisällyttämisen mahdollisuutta haihdutusprosessiin arvioitiin. Kiehumispisteen kohoama vähäisissä vesipitoisuuksissa on nopeaa, mikä vaikeuttaa monivaihelauhdutuksen käyttämistä. Oikeilla paineilla ja lämpötiloilla se saatiin kuitenkin sisällytettyä, jos viimeisen haihduttimen lämmittämiseen käytettiin ulkoista energiaa. Monivaihelauhdutus säästi selvästi lämmitysenergiaa sekä auttoi kierrätystä jakamalla höyryn suureen virtaan erittäin puhdasta vesihöyryä ja pienempään virtaan, jossa lähes kaikki höyrystynyt ioninen neste oli

Rigidity of generic singularities of mean curvature flow

Shrinkers are special solutions of mean curvature flow (MCF) that evolve by rescaling and model the singularities. While there are infinitely many in each dimension, [CM1] showed that the only generic are round cylinders \SS^k\times \RR^{n-k}. We prove here that round cylinders are rigid in a very strong sense. Namely, any other shrinker that is sufficiently close to one of them on a large, but compact, set must itself be a round cylinder. To our knowledge, this is the first general rigidity theorem for singularities of a nonlinear geometric flow. We expect that the techniques and ideas developed here have applications to other flows. Our results hold in all dimensions and do not require any a priori smoothness.Comment: revised after acceptance for Publications IHE

Topological Change in Mean Convex Mean Curvature Flow

Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy group of the complementary region can die only if there is a shrinking S^k x R^(n-k) singularity for some k less than or equal to m. We also prove that for each m from 1 to n, there is a nonempty open set of compact, mean convex regions K in R^(n+1) with smooth boundary for which the resulting mean curvature flow has a shrinking S^m x R^(n-m) singularity.Comment: 19 pages. This version includes a new section proving that certain kinds of mean curvature flow singularities persist under arbitrary small perturbations of the initial surface. Newest update (Oct 2013) fixes some bibliographic reference

The round sphere minimizes entropy among closed self-shrinkers

The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy of the initial hypersurface bounds the entropy at all future singularities. We show here that not only does the round sphere have the lowest entropy of any closed singularity, but there is a gap to the second lowest

Higher regularity of the inverse mean curvature flow

We prove higher regularity properties of inverse mean curvature flow in Euclidean space: A sharp lower bound for the mean curvature is derived for star-shaped surfaces, independently of the initial mean curvature. It is also shown that solutions to the inverse mean curvature flow are smooth if the mean curvature is bounded from below. As a consequence we show that weak solutions of the inverse mean curvature flow are smooth for large times, beginning from the first time where a surface in the evolution is star-shaped

The mean curvature at the first singular time of the mean curvature flow

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)$, for $t\in [0,T)$. We prove that the mean curvature blows up at the first singular time $T$ if all singularities are of type I. In the case $n = 2$, regardless of the type of a possibly forming singularity, we show that at the first singular time the mean curvature necessarily blows up provided that either the Multiplicity One Conjecture holds or the Gaussian density is less than two. We also establish and give several applications of a local regularity theorem which is a parabolic analogue of Choi-Schoen estimate for minimal submanifolds