Showing papers in "Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze in 2012"
TL;DR: The Ricci almost soliton as discussed by the authors is a natural extension of the concept of gradient Ricci soliton, and it has been shown to be a generalization of the Ricci Almost Soliton.
Abstract: We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geometric quantities are derived. Some basic tools from the weighted manifold theory such as general weighted volume comparisons and maximum principles at infinity for diffusion operators are discussed. Mathematics Subject Classification (2010): 53C21.
191 citations
TL;DR: In this article, the authors consider functions that satisfy the identity ue(x) = α 2 { sup Be(x ue + inf Be (x) ue } + β ∫.
Abstract: We consider functions that satisfy the identity ue(x) = α 2 { sup Be(x) ue + inf Be(x) ue } + β ∫
129 citations
TL;DR: In this paper, the Cauchy problem of the Novikov equation was studied and it was shown that peakon solutions to the equation are global weak solutions, and that the equation has smooth solutions which exist globally in time.
Abstract: In this paper, we mainly study the Cauchy problem of the Novikov equation. We first establish the local well-posedness and give the precise blow-up scenario for the equation. Then we show that the equation has smooth solutions which exist globally in time. Finally we prove that peakon solutions to the equation are global weak solutions.
105 citations
TL;DR: In this paper, the authors consider nonlinear Neumann problems driven by the p-Laplacian plus an indefinite potential, and develop the spectral properties of such differential operators and prove existence and multiplicity theorems for resonant problems.
Abstract: We consider nonlinear Neumann problems driven by the p-Laplacian plus an indefinite potential. First we develop the spectral properties of such differential operators. Subsequently, using these spectral properties and variational methods based on critical point theory, truncation techniques and Morse theory, we prove existence and multiplicity theorems for resonant problems.
78 citations
TL;DR: In this article, the authors analyse conditions for an evolution equation with a drift and fractional diffusion to have a Holder continuous solution and obtain Holder estimates for the solution for any bounded drift.
Abstract: We analyse conditions for an evolution equation with a drift and fractional diffusion to have a Holder continuous solution. In case the diffusion is of order one or more, we obtain Holder estimates for the solution for any bounded drift. In the case when the diffusion is of order less than one, we require the drift to be a Holder continuous vector field in order to obtain the same type of regularity result.
62 citations
TL;DR: In this paper, it was shown that the operator L = (1+|x| � )� admits realizations generating contraction or analytic semigroups in L p (R N ).
Abstract: In this paper we prove that, under suitable assumptions on � > 0, the operator L = (1+|x| � )� admits realizations generating contraction or analytic semigroups in L p (R N ). For some values of �, we also explicitly characterize the domain of L. Finally, some informations about the location and composition of the spectrum are given.
55 citations
TL;DR: Weproveshorttimeexistence, uniqueness andcontinuous dependence on the initial data of smooth solutions of quasilinear locally parabolic equations of arbitrary even order on closed manifolds are studied in this paper.
Abstract: Weproveshorttimeexistence, uniquenessandcontinuousdependence on the initial data of smooth solutions of quasilinear locally parabolic equations of arbitrary even order on closed manifolds.
39 citations
TL;DR: In this paper, it was shown that any torus in R 3 with energy at most 8π − δ has a representative under the Mobius action for which the induced metric and a conformal metric of constant (zero) curvature are uniformly equivalent, with constants depending only on δ > 0.
Abstract: The Willmore energy of a closed surface in Rn is the integral of its squared mean curvature, and is invariant under Mobius transformations of Rn . We show that any torus in R3 with energy at most 8π − δ has a representative under the Mobius action for which the induced metric and a conformal metric of constant (zero) curvature are uniformly equivalent, with constants depending only on δ > 0. An analogous estimate is also obtained for closed, orientable surfaces of fixed genus p ≥ 1 in R3 or R4, assuming suitable energy bounds which are sharp for n = 3. Moreover, the conformal type is controlled in terms of the energy bounds. Mathematics Subject Classification (2010): 53A05 (primary); 53A30, 53C21, 49Q15 (secondary).
36 citations
TL;DR: In this paper, it was shown that intrinsic Lipschitz domains have locally finite perimeter, and the existence of a boundary trace operator for functions with bounded variation on these domains was shown.
Abstract: WeintroduceintrinsicLipschitzhypersurfacesinCarnot-Carath´ eodory spaces and prove that intrinsic Lipschitz domains have locally finite perimeter. We also show the existence of a boundary trace operator for functions with bounded variation on Lipschitz domains and obtain extension results for such functions. In particular, we characterize their trace space.
33 citations
TL;DR: In this article, the best constant in the L2 Folland-Stein inequality on the quaternionic Heisenberg group and non-negative functions for which equality holds was determined.
Abstract: We determine the best (optimal) constant in the L2 Folland-Stein inequality on the quaternionic Heisenberg group and the non-negative functions for which equality holds
29 citations
TL;DR: In this article, it was shown that under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected, and several examples of arrangements with eight, nine and ten lines were shown to have disconnected realization spaces.
Abstract: We prove that, under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected. We also give several examples of arrangements with eight, nine and ten lines that have disconnected realization spaces.
TL;DR: In this article, a new proof for the superigidity of lattices in solvable Lie groups was obtained, which can be applied to compute the Betti numbers of a compact solvmanifold G/! even in the case that the solvable lie group G and the lattice! do not satisfy the Mostow condition.
Abstract: Using results by D.Witte [35] on the superigidity of lattices in solvable Lie groups we get a new proof of a recent remarkable result obtained by D. Guan [15] on the de Rham cohomology of a compact solvmanifold, i.e., of a quotient of a connected and simply connected solvable Lie group G by a lattice !. This result can be applied to compute the Betti numbers of a compact solvmanifold G/! even in the case that the solvable Lie group G and the lattice ! do not satisfy the Mostow condition. Mathematics Subject Classification (2010): 53C30 (primary); 22E25, 22E40 (secondary).
TL;DR: In this article, the authors present conformal structures in signature (3, 2) for which the Fefferman-Graham ambient metric is equal to the non-compact G2(2) Lie group.
Abstract: We present conformal structures in signature (3, 2) for which the
holonomy of the Fefferman-Graham ambient metric is equal to the non-compact
exceptional Lie group G2(2). We write down the resulting 8-parameter family of
G2(2)-metrics in dimension seven explicitly in an appropriately chosen coordinate
system on the ambient space.
TL;DR: In this article, it was shown that S5 is smooth except possibly at isolated points, where the tangent connections between S5 and S5 are smooth except for isolated points.
Abstract: Special Legendrian Integral Cycles in S5 are the links of the tangent
cones to Special Lagrangian integer multiplicity rectifiable currents in Calabi-
Yau 3-folds. We show that Special Legendrian Cycles are smooth except possibly
at isolated points.
TL;DR: In this article, the compactness of resolvent of the Fokker-Planck operator with potential V(x) was analyzed and some kind of conditions imposed on the potential to ensure the validity of global hypoelliptic estimates.
Abstract: Inthis paper westudytheFokker-Planckoperator withpotential V(x), and analyze some kind of conditions imposed on the potential to ensure the validity of global hypoelliptic estimates (see Theorem 1.1). As a consequence, we obtain the compactness of resolvent of the Fokker-Planck operator if either the Witten Laplacian on 0-forms has a compact resolvent or some additional assumption on the behavior of the potential at infinity is fulfilled. This work improves the previous results of H´ erau-Nier [5] and Helffer-Nier [3], by obtaining a better global hypoelliptic estimate under weaker assumptions on the potential.
TL;DR: In this article, the generalized boundary value problem for nonnegative solutions of of $- Δ u+g(u)=0$ in a bounded Lipschitz domain was studied and a trace in the class of outer regular Borel measures was defined.
Abstract: We study the generalized boundary value problem for nonnegative solutions of of $-\Delta u+g(u)=0$ in a bounded Lipschitz domain $\Omega$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Omega$, we define a trace in the class of outer regular Borel measures. We amphasize the case where $g(u)=|u|^{q-1}u$, $q>1$. When $\Omega$ is (locally) a cone with vertex $y$, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that $\Omega$ possesses a tangent cone at every boundary point and $q$ is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace.
TL;DR: In this paper, the authors extended some previous results about the Cauchy problem and the quasi-stationary limit to the case where the magnetic permeability and the electric permittivity are variable.
Abstract: In this paper we continue the investigation of the Maxwell-LandauLifschitz and Maxwell-Bloch equations. In particular we extend some previous results about the Cauchy problem and the quasi-stationary limit to the case where the magnetic permeability and the electric permittivity are variable.
TL;DR: In this article, the authors revisited a homogenization problem studied by L. Tartar related to a tridimensional Stokes equation perturbed by a drift (connected to the Coriolis force).
Abstract: This paper revisits a homogenization problem studied by L. Tartar related to a tridimensional Stokes equation perturbed by a drift (connected to the Coriolis force). Here, a scalar equation and a two-dimensional Stokes equation with a $L^2$-bounded oscillating drift are considered. Under higher integrability conditions the Tartar approach based on the oscillations test functions method applies and leads to a limit equation with an extra zero-order term. When the drift is only assumed to be equi-integrable in $L^2$, the same limit behaviour is obtained. However, the lack of integrability makes difficult the direct use of the Tartar method. A new method in the context of homogenization theory is proposed. It is based on a parametrix of the Laplace operator which permits to write the solution of the equation as a solution of a fixed point problem, and to use truncated functions even in the vector-valued case. On the other hand, two counter-examples which induce different homogenized zero-order terms actually show the optimality of the equi-integrability assumption.
TL;DR: In this article, a priori estimates for the resolvent of Navier equation in elasticity, when one approaches the spectrum (Limiting Absorption Principles), were presented.
Abstract: We prove some a priori estimates for the resolvent of Navier equation in elasticity, when one approaches the spectrum (Limiting Absorption Principles). They are extensions of analogous estimates for the resolvent of the euclidean Laplacian in Rn. As a consequence, we get some results for the evolution equation and for the spectral measure
TL;DR: In this paper, the authors show that quasi-K´ahler Chern-flat almost Hermitian structures on compact manifolds are in correspondence to complex parallelisable Hermitians satisfying the second Gray identity.
Abstract: The study of quasi-K¨ahler Chern-flat almost Hermitian manifolds is
strictly related to the study of anti-bi-invariant almost complex Lie algebras. In
the present paper we show that quasi-K¨ahler Chern-flat almost Hermitian structures
on compact manifolds are in correspondence to complex parallelisable Hermitian
structures satisfying the second Gray identity. From an algebraic point
of view this correspondence reads as a natural correspondence between anti-biinvariant
almost complex structures on Lie algebras and bi-invariant complex
structures. Some natural algebraic problems are approached and some exotic
examples are carefully described.
TL;DR: In this paper, two nonnegative solutions of the Dirichlet boundary condition were found, where 0 0 is large enough to give rise to two distinct nonnegative solution of the original problem.
Abstract: We find two nontrivial solutions of the equation −!u = (− 1 uβ + λu )χ{u>0} in % with Dirichlet boundary condition, where 0 0 is large enough, we find two critical points of the corresponding e-functional which, at the limit as e→ 0, give rise to two distinct nonnegative solutions of the original problem. Another approach is based on perturbations of the domain %, we then find a unique positive solution for λ large enough. We derive gradient estimates to guarantee convergence of approximate solutions ue to a true solution u of the problem. Mathematics Subject Classification (2010): 34B16 (primary); 35J20, 35B65 (secondary).
TL;DR: In this paper, it was shown that the twisted Alexander polynomial of a torus knot with irreducible SL(2,C)-representation is locally constant and gave an explicit formula for it.
Abstract: We prove that the twisted Alexander polynomial of a torus knot with
an irreducible SL(2,C)-representation is locally constant. In the case of a (2, q)
torus knot, we can give an explicit formula for the twisted Alexander polynomial
and deduce Hirasawa-Murasugi’s formula for the total twisted Alexander polynomial.
We also give examples which address a mis-statement in a paper of Silver
and Williams.
TL;DR: In this article, Fourier transform estimates for functions from certain classes under the Muckenhoupt conditions were studied and the sharpness of these conditions was proved for function classes under Ap or A2p.
Abstract: Fourier transform estimates for ‖ f ‖Lq,w via ‖ f ‖L p,w from above and from below are studied. For p = q, equivalence results, i.e., C1‖ f ‖L p,w ≤ ‖ f ‖L p,w ≤ C2‖ f ‖L p,w , w(x) = w(1/x)x p−2, 1 ≤ p <∞, are shown to be valid for functions from certain classes under the Muckenhoupt conditions: w ∈ Ap or w ∈ A2p . Sharpness of these conditions is proved. Mathematics Subject Classification (2010): 42A38 (primary); 26D15, 46E30 (secondary).
TL;DR: In this article, it was shown that the modified Calabi flow converges to an extremal metric near a given extremal distance in an exponential time when the initial data is invariant under the maximal compact subgroup of the identity component of the reduced automorphism group.
Abstract: We prove that on a K\\\"ahler manifold admitting an extremal metric $\\omega$ and for any K\\\"ahler potential $\\varphi_0$ close to $\\omega$, the Calabi flow starting at $\\varphi_0$ exists for all time and the modified Calabi flow starting at $\\varphi_0$ will always be close to $\\omega$. Furthermore, when the initial data is invariant under the maximal compact subgroup of the identity component of the reduced automorphism group, the modified Calabi flow converges to an extremal metric near $\\omega$ exponentially fast.
TL;DR: In this paper, the authors construct one-ended constant mean curvature surfaces with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic.
Abstract: We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces cannot exist in Euclidean space, but we show that the gradient of the ambient scalar curvature acts as a ‘friction term’ which permits the usual analytic gluing construction to be carried out.
TL;DR: In this article, an abstract framework for John-Nirenberg inequalities associated to BMO-type spaces is developed, which can be seen as the sequel of [6], where the authors introduced a very general framework for atomic and molecular Hardy spaces.
Abstract: In this paper, we develop an abstract framework for John-Nirenberg inequalities associated to BMO-type spaces. This work can be seen as the sequel of [6], where the authors introduced a very general framework for atomic and molecular Hardy spaces. Moreover, we show that our assumptions allow us to recover some already known John-Nirenberg inequalities. We give applications to the atomic Hardy spaces too.
TL;DR: It is proved that a complex surface S with irregularity q(S)=5 that has no irrational pencil of genus >1 has geometric genus p_g(S)>7 and one is able to classify minimal surfaces S of general type with q( S)=5 and p-3 that have no rationality of genus>1 and with the lowest possible geometric genusp_g=2q-3.
Abstract: We prove that a complex surface S with irregularity q(S) = 5 that has no irrational pencil of genus > 1 has geometric genus pg(S) � 8. As a consequence, one is able to classify minimal surfaces S of general type with q(S) = 5 and pg(S) 1 and with the lowest possible geometric genus pg = 2q 3. This gives some evidence for the conjecture that the only irregular surface with no irrational pencil of genus > 1 and pg = 2q 3 is the symmetric product of a genus three curve.
TL;DR: In this article, the authors characterized closed $2k$-manifolds admitting smooth maps into $(k+1)$-mansifolds with only finitely many critical points, for $k\in\{2,4\}$.
Abstract: We characterize those closed $2k$-manifolds admitting smooth maps into $(k+1)$-manifolds with only finitely many critical points, for $k\in\{2,4\}$. We compute then the minimal number of critical points of such smooth maps for $k=2$ and, under some fundamental group restrictions, also for $k=4$. The main ingredients are King's local classification of isolated singularities, decomposition theory, low dimensional cobordisms of spherical fibrations and 3-manifolds topology.
TL;DR: The theory of metric currents of Ambrosio and Kirchheim in the setting of spaces admitting differentiable structures in the sense of Cheeger and Keith was examined in this article.
Abstract: We examine the theory of metric currents of Ambrosio and Kirchheim in the setting of spaces admitting differentiable structures in the sense of Cheeger and Keith. We prove that metric forms which vanish in the sense of Cheeger on a set must also vanish when paired with currents concentrated along that set. From this we deduce a generalization of the chain rule, and show that currents of absolutely continuous mass are given by integration against measurable k-vector fields. We further prove that if the underlying metric space is a Carnot group with its Carnot-Caratheodory distance, then every metric current T satisfies T !θ= 0 and T !dθ= 0, whenever θ ∈ "1(G) annihilates the horizontal bundle of G. Moreover, this condition is necessary and sufficient for a metric current with respect to the Riemannian metric to extend to one with respect to the Carnot-Caratheodory metric, provided the current either is locally normal, or has absolutely continuous mass. Mathematics Subject Classification (2010): 30L99 (primary); 49Q15 (secondary).
TL;DR: In this paper, the authors prove the existence and uniqueness of a quasivariational sweeping process on functions of bounded variation and show that the condition on the jump size can be replaced by suitable conditions on the shape of the convex set.
Abstract: We prove existence and uniqueness of a quasivariational sweeping
process on functions of bounded variation thereby generalizing previous results
for absolutely continuous functions. It turns out that the size of the discontinuities
plays a crucial role: In case they are small enough we prove existence and
uniqueness. For large jumps we present a counterexample to the uniqueness of
the solution. Finally we show that the condition on the jump size can be replaced
by suitable conditions on the shape of the convex set.