CVGMT Papershttp://cvgmt.sns.it/papers/en-usWed, 20 Jun 2018 15:18:18 +0000$C^{1,α}$-Regularity for variational problems in the Heisenberg grouphttp://cvgmt.sns.it/paper/3927/S. Mukherjee, X. Zhong.<p> We study the regularity of minima of scalar variational integrals of$p$-growth, $1<p<\infty$, in the Heisenberg group and prove the H\"oldercontinuity of horizontal gradient of minima.</p>http://cvgmt.sns.it/paper/3927/Modulus of families of sets of finite perimeter and quasiconformal maps between metric spaces of globally $Q$-bounded geometryhttp://cvgmt.sns.it/paper/3926/R. Jones, P. Lahti, N. Shanmugalingam.<p>We generalize a result of Kelly to the setting of Ahlfors $Q$-regular metric measure spacessupporting a $1$-Poincar\'e inequality. It is shown that if $X$ and $Y$ are two Ahlfors $Q$-regular spaces supportinga $1$-Poincar\'e inequality and $f:X\to Y$ is a quasiconformal mapping, then the $Q/(Q-1)$-modulus of the collectionof measures $\mathcal{H}^{Q-1}\vert_{\Sigma E}$ corresponding to any collection of sets $E\subset X$ of finite perimeteris quasi-preserved by $f$. We also show that for $Q/(Q-1)$-modulus almost every $\Sigma E$,if the image surface $\Sigma f(E)$ does not see the singular set of $f$ as a large set, then $f(E)$ is also of finite perimeter. Even in the standard Euclidean setting our results are more general than that of Kelly, and hence are new even in there.</p>http://cvgmt.sns.it/paper/3926/A variational characterisation of the second eigenvalue of the p-Laplacian on quasi open setshttp://cvgmt.sns.it/paper/3925/N. Fusco, S. Mukherjee, Y. R. Y. Zhang.<p>In this article, we prove a minimax characterisation of the second eigenvalue of the p- Laplacian operator on p-quasi open sets, using a construction based on minimizing movements. This leads also to an existence theorem for spectral functionals depending on the first two eigenvalues of the p-Laplacian.</p>http://cvgmt.sns.it/paper/3925/Homogenization of high-contrast Mumford-Shah energieshttp://cvgmt.sns.it/paper/3924/X. Pellet, L. Scardia, C. I. Zeppieri.<p>We prove a homogenization result for Mumford-Shah-type energies associated to a brittle composite material with weak inclusions distributed periodically at a scale $\varepsilon>0$. The matrix and the inclusions in the material have the same elastic moduli but very different toughness moduli, with the ratio of the toughness modulus in the matrix and in the inclusions being $1/\beta_\varepsilon$, with $\beta_{\varepsilon}>0$ small. We show that the high-contrast behavior of the composite leads to the emergence of interesting effects in the limit: The volume and surface energy densities interact by $\Gamma$-convergence, and the limit volume energy is not a quadratic form in the critical scaling $\beta_\varepsilon = \varepsilon$, unlike the $\varepsilon$-energies, and unlike the extremal limit cases.</p>http://cvgmt.sns.it/paper/3924/Quantitative analysis of finite-difference approximations of free-discontinuity problemshttp://cvgmt.sns.it/paper/3923/A. Bach, A. Braides, C. I. Zeppieri.<p>Motivated by applications to image reconstruction, in this paper we analyse a finite-difference discretisation of the Ambrosio-Tortorelli functional. Denoted by $\varepsilon$ the elliptic-approximation parameter and by $\delta$ the discretisation step-size, we fully describe the relative impact of $\varepsilon$ and $\delta$ in terms of $\Gamma$-limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when $\varepsilon$ and $\delta$ are of the same order, the underlying lattice structure affects the $\Gamma$-limit which turns out to be an anisotropic free-discontinuity functional.</p>http://cvgmt.sns.it/paper/3923/Periodic homogenization of nonlocal Hamilton-Jacobi equations with coercive gradietn termshttp://cvgmt.sns.it/paper/3922/M. Bardi, A. Cesaroni, E. Topp.<p>This paper deals with the periodic homogenization of nonlocal parabolic Hamilton-Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different limit problems.</p>http://cvgmt.sns.it/paper/3922/Dynamic perfect plasticity and damage in viscoelastic solidshttp://cvgmt.sns.it/paper/3921/E. Davoli, T. Roubicek, U. Stefanelli.<p>In this paper we analyze an isothermal and isotropic model for viscoelastic media combining linearized perfect plasticity (allowing for concentration of plastic strain and development of shear bands) and damage effects in a dynamic setting. The interplay between the viscoelastic rheology with inertia, elasto-plasticity, and unidirectional rate-dependent incomplete damage affecting both the elastic and viscous response, as well as the plastic yield stress, is rigorously characterized by showing existence of weak solutions to the constitutive and balance equations of the model. The analysis relies on the notions of plastic-strain measures and bounded-deformation displacements, on sophisticated time-regularity estimates to establish a duality between acceleration and velocity of the elastic displacement, on the theory of rate-independent processes for the energy conservation in the dynamical-plastic part, and on the proof of the strong convergence of the elastic strains. Existence of a suitably defined weak solutions is proved rather constructively by using a staggered two-step time discretization scheme.</p>http://cvgmt.sns.it/paper/3921/Fattening and nonfattening phenomena for planar nonlocal curvature flowshttp://cvgmt.sns.it/paper/3920/A. Cesaroni, S. Dipierro, M. Novaga, E. Valdinoci.<p>We discuss fattening phenomenon for the evolution of planar curves according to their nonlocal curvature. More precisely,we consider a class of generalized curvatures which correspond to the first variation of suitable nonlocal perimeter functionals, defined in terms of an interaction kernel $K$,which is symmetric, nonnegative,possibly singular at the origin, and satisfies appropriate integrability conditions. </p><p>We prove a general result about uniqueness of the geometric evolutionsstarting from regular sets with positive $K$-curvatureand we discuss the fattening phenomenon for the evolutionstarting from the cross, showing that this phenomenon is very sensitiveto the strength of the interactions. As a matter of fact, we show that the fattening of the cross occursfor kernels with sufficiently large mass near the origin,while for kernels that are sufficiently weak near the originsuch a fattening phenomenon does not occur.</p><p>We also provide some further results in the case of the fractionalmean curvature flow, showingthat strictly starshaped sets have a unique geometric evolution.</p><p>Moreover, we exhibit two illustrative examples of closed nonregular curves,the first with a Lipschitz-type singularity and the second with a cusp-typesingularity, given by two tangent circles of equal radius,whose evolution develops fattening in the first case,and is uniquely defined in the second, thus remarkingthe high sensitivity of the fattening phenomenonin terms of the regularity of the initial datum.The latter example is in striking contrast to the classical case of the (local) curvature flow,where two tangent circles always develop fattening.</p><p>As a byproduct of our analysis, we providealso a simple proof of the fact that the cross is nota $K$-minimal set for the nonlocal perimeter functional associated to $K$.</p>http://cvgmt.sns.it/paper/3920/Monotone Volume Formulas for Geometric Flowshttp://cvgmt.sns.it/paper/3918/R. Buzano.<p>We consider a closed manifold M with a Riemannian metric g(t) evolving indirection -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We provethat if S satisfies a certain tensor inequality, then one can construct aforwards and a backwards reduced volume quantity, the former beingnon-increasing, the latter being non-decreasing along the flow. In the casewhere S=Ric is the Ricci curvature of M, the result corresponds to Perelman'swell-known reduced volume monotonicity for the Ricci flow. Some other examplesare given in the second section of this article, the main examples andmotivation for this work being List's extended Ricci flow system, the Ricciflow coupled with harmonic map heat flow and the mean curvature flow inLorentzian manifolds with nonnegative sectional curvatures. With our approach,we find new monotonicity formulas for these flows.</p><p>Published under previous name Reto Müller (please cite as Müller). Name changed here in order to import to author page correctly.</p>http://cvgmt.sns.it/paper/3918/Ricci flow coupled with harmonic map flowhttp://cvgmt.sns.it/paper/3917/R. Buzano.<p>We investigate a new geometric flow which consists of a coupled system of theRicci flow on a closed manifold M with the harmonic map flow of a map phi fromM to some closed target manifold N with a (possibly time-dependent) positivecoupling constant alpha. This system can be interpreted as the gradient flow ofan energy functional F<sub>alpha</sub> which is a modification of Perelman's energy F forthe Ricci flow, including the Dirichlet energy for the map phi. Surprisingly,the coupled system may be less singular than the Ricci flow or the harmonic mapflow alone. In particular, we can always rule out energy concentration of phia-priori - without any assumptions on the curvature of the target manifold N -by choosing alpha large enough. Moreover, if alpha is bounded away from zero itsuffices to bound the curvature of (M,g(t)) to also obtain control of phi andall its derivatives - a result which is clearly not true for alpha = 0. Besidesthese new phenomena, the flow shares many good properties with the Ricci flow.In particular, we can derive the monotonicity of an entropy functional W<sub>alpha</sub>similar to Perelman's Ricci flow entropy W and of so-called reduced volumefunctionals. We then apply these monotonicity results to rule out non-trivialbreathers and geometric collapsing at finite times.</p><p>Published under previous name Reto Müller (please cite as Müller). Name changed here in order to import to author page correctly.</p>http://cvgmt.sns.it/paper/3917/On Type I Singularities in Ricci flowhttp://cvgmt.sns.it/paper/3916/R. Buzano, J. Enders, P. Topping.<p>We define several notions of singular set for Type I Ricci flows and showthat they all coincide. In order to do this, we prove that blow-ups aroundsingular points converge to nontrivial gradient shrinking solitons, thusextending work of Naber. As a by-product we conclude that the volume of afinite-volume singular set vanishes at the singular time. We also define anotion of density for Type I Ricci flows and use it to prove a regularitytheorem reminiscent of White's partial regularity result for mean curvatureflow.</p><p>Note name change of one author from Reto Müller to Reto Buzano in 2015. Please cite as Enders-Müller-Topping. Name changed here in order to import to author page correctly.</p>http://cvgmt.sns.it/paper/3916/Perelman's Entropy Functional at Type I Singularities of the Ricci Flowhttp://cvgmt.sns.it/paper/3914/R. Buzano, C. Mantegazza.<p>We study blow-ups around fixed points at Type I singularities of the Ricciflow on closed manifolds using Perelman's W-functional. First, we give analternative proof of the result obtained by Naber and Enders-M\"{u}ller-Toppingthat blow-up limits are non-flat gradient shrinking Ricci solitons. Our secondand main result relates a limit W-density at a Type I singular point to theentropy of the limit gradient shrinking soliton obtained by blowing-up at thispoint. In particular, we show that no entropy is lost at infinity during theblow-up process.</p><p>Note name change of one author from Reto Müller to Reto Buzano in 2015. Please cite as Mantegazza-Müller. Name changed here in order to import to author page correctly.</p>http://cvgmt.sns.it/paper/3914/A compactness theorem for complete Ricci shrinkershttp://cvgmt.sns.it/paper/3915/R. Buzano, R. Haslhofer.<p>We prove precompactness in an orbifold Cheeger-Gromov sense of completegradient Ricci shrinkers with a lower bound on their entropy and a localintegral Riemann bound. We do not need any pointwise curvature assumptions,volume or diameter bounds. In dimension four, under a technical assumption, wecan replace the local integral Riemann bound by an upper bound for the Eulercharacteristic. The proof relies on a Gauss-Bonnet with cutoff argument.</p><p>Note name change of one author from Reto Müller to Reto Buzano in 2015. Please cite as Haslhofer-Müller. Name changed here in order to import to author page correctly.</p>http://cvgmt.sns.it/paper/3915/Dynamical stability and instability of Ricci-flat metricshttp://cvgmt.sns.it/paper/3913/R. Buzano, R. Haslhofer.<p>In this short article, we improve the dynamical stability and instabilityresults for Ricci-flat metrics under Ricci flow proved by Sesum and Haslhofer,getting rid of the integrability assumption.</p><p>Note name change of one author from Reto Müller to Reto Buzano in 2015. Please cite as Haslhofer-Müller. Name changed here in order to import to author page correctly.</p>http://cvgmt.sns.it/paper/3913/A note on the compactness theorem for 4d Ricci shrinkershttp://cvgmt.sns.it/paper/3912/R. Buzano, R. Haslhofer.<p>In arXiv:1005.3255 we proved an orbifold Cheeger-Gromov compactness theoremfor complete 4d Ricci shrinkers with a lower bound for the entropy, an upperbound for the Euler characterisic, and a lower bound for the gradient of thepotential at large distances. In this note, we show that the last twoassumptions in fact can be removed. The key ingredient is a recent estimate ofCheeger-Naber arXiv:1406.6534.</p><p>Note name change of one author from Reto Müller to Reto Buzano in 2015. Please cite as Haslhofer-Müller. Name changed here in order to import to author page correctly.</p>http://cvgmt.sns.it/paper/3912/Smooth long-time existence of Harmonic Ricci Flow on surfaceshttp://cvgmt.sns.it/paper/3903/R. Buzano, M. Rupflin.<p> We prove that at a finite singular time for the Harmonic Ricci Flow on asurface of positive genus both the energy density of the map component and thecurvature of the domain manifold have to blow up simultaneously. As animmediate consequence, we obtain smooth long-time existence for the HarmonicRicci Flow with large coupling constant.</p>http://cvgmt.sns.it/paper/3903/The Chern-Gauss-Bonnet formula for singular non-compact four-dimensional manifoldshttp://cvgmt.sns.it/paper/3904/R. Buzano, H. T. Nguyen.<p> We generalise the classical Chern-Gauss-Bonnet formula to a class of4-dimensional manifolds with finitely many conformally flat ends and singularpoints. This extends results of Chang-Qing-Yang in the smooth case. Under theassumptions of finite total Q curvature and positive scalar curvature at theends and at the singularities, we obtain a new Chern-Gauss-Bonnet formula witherror terms that can be expressed as isoperimetric deficits. This is the firstsuch formula in a dimension higher than two which allows the underlyingmanifold to have isolated branch points or conical singularities.</p>http://cvgmt.sns.it/paper/3904/The moduli space of two-convex embedded sphereshttp://cvgmt.sns.it/paper/3901/R. Buzano, R. Haslhofer, O. Hershkovits.<p> We prove that the moduli space of 2-convex embedded n-spheres in R<sup>{n+1}</sup> ispath-connected for every n. Our proof uses mean curvature flow with surgery andcan be seen as an extrinsic analog to Marques' influential proof of thepath-connectedness of the moduli space of positive scalar curvature metics onthree-manifolds.</p>http://cvgmt.sns.it/paper/3901/Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and areahttp://cvgmt.sns.it/paper/3902/R. Buzano, B. Sharp.<p> We prove qualitative estimates on the total curvature of closed minimalhypersurfaces in closed Riemannian manifolds in terms of their index and area,restricting to the case where the hypersurface has dimension less than seven.In particular, we prove that if we are given a sequence of closed minimalhypersurfaces of bounded area and index, the total curvature along the sequenceis quantised in terms of the total curvature of some limit surface, plus a sumof total curvatures of complete properly embedded minimal hypersurfaces inEuclidean space - all of which are finite. Thus, we obtain qualitative controlon the topology of minimal hypersurfaces in terms of index and area as acorollary.</p>http://cvgmt.sns.it/paper/3902/The higher-dimensional Chern-Gauss-Bonnet formula for singular conformally flat manifoldshttp://cvgmt.sns.it/paper/3899/R. Buzano, H. T. Nguyen.<p> In a previous article, we generalised the classical four-dimensionalChern-Gauss-Bonnet formula to a class of manifolds with finitely manyconformally flat ends and singular points, in particular obtaining the firstsuch formula in a dimension higher than two which allows the underlyingmanifold to have isolated conical singularities. In the present article, weextend this result to all even dimensions $n\geq 4$ in the case of a class ofconformally flat manifolds.</p>http://cvgmt.sns.it/paper/3899/