The Journal of Geometric Analysis最新文献

We prove a logarithmic Sobolev inequality along the (textrm{G}_{2})-Laplacian flow. A uniform Sololev inequality along the (textrm{G}_{2})-Laplacian flow with uniformly bounded scalar curvature is derived from the logarithmic Sobolev inequality. The uniform Sololev inequality implies a (kappa )-noncollapsing estimate for the (textrm{G}_{2})-Laplacian flow with uniformly bounded scalar curvature. Furthermore, by using the logarithmic Sobolev inequality, we prove Gaussian-type upper bounds for the heat kernel along the (textrm{G}_{2})-Laplacian flow with uniformly bounded scalar curvature.

In this paper we study the stability of a Killing cylinder in hyperbolic 3-space when regarded as a capillary surface for the partitioning problem. In contrast with the Euclidean case, we consider a variety of totally umbilical support surfaces, including horospheres, totally geodesic planes, equidistant surfaces and round spheres. In all of them, we explicitly compute the Morse index of the corresponding eigenvalue problem for the Jacobi operator. We also address the stability of compact pieces of Killing cylinders with Dirichlet boundary conditions when the boundary is formed by two fixed circles, exhibiting an analogous to the Plateau–Rayleigh instability criterion for Killing cylinders in the Euclidean space. Finally, we prove that the Delaunay surfaces can be obtained by bifurcating Killing cylinders supported on geodesic planes.

We study a ramification of a phenomenon discovered by Baird and Eells (in: Looijenga et al (eds) Geometry Symposium Utrecht 1980. Lecture Notes in Mathematics, Springer, Berlin, 1981) i.e. that non-constant harmonic morphisms (Phi : {{mathfrak {M}}}^{textrm{N}} rightarrow N^2) from a (mathrm N)-dimensional ((textrm{N} ge 3)) Riemannian manifold ({{mathfrak {M}}}^{textrm{N}}), into a Riemann surface (N^2), can be characterized as those horizontally weakly conformal maps having minimal fibres. We recover Baird–Eells’ result for (S^1) invariant harmonic morphisms (Phi : {{mathfrak {M}}}^{2n+2} rightarrow N^2) from a class of Lorentzian manifolds arising as total spaces ({{mathfrak {M}}} = C(M)) of canonical circle bundles (S^1 rightarrow {{mathfrak {M}}} rightarrow M) over strictly pseudoconvex CR manifolds (M^{2n+1}). The corresponding base maps (phi : M^{2n+1} rightarrow N^2) are shown to satisfy (lim _{epsilon rightarrow 0^+} , pi _{{{mathscr {H}}}^phi } , mu ^{{{mathscr {V}}}^phi }_epsilon = 0), where (mu ^{{{mathscr {V}}}^phi }_epsilon ) is the mean curvature vector of the vertical distribution ({{mathscr {V}}}^phi = textrm{Ker} (d phi )) on the Riemannian manifold ((M, , g_epsilon )), and ({ g_epsilon }_{0< epsilon < 1}) is a family of contractions of the Levi form of the pseudohermitian manifold ((M, , theta )).

The purpose of this article is to study rigidity of free boundary minimal two-disks that locally maximize the modified Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature and mean convex boundary. Assuming the strict stability of (Sigma ), we prove that a neighborhood of it in M is isometric to one of the half de Sitter–Schwarzschild space.

We study the chordal Loewner equation associated with certain driving functions that produce infinitely many slits. Specifically, for a choice of a sequence of positive numbers ((b_n)_{nge 1}) and points of the real line ((k_n)_{nge 1}), we explicitily solve the Loewner PDE

in (mathbb {H}times [0,1)). Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as (trightarrow 1^-).

In this paper, we prove weighted (L^p) estimates for the canonical solutions on product domains. As an application, we show that if (pin [4, infty )), the (bar{partial }) equation on the Hartogs triangle with (L^p) data admits (L^p) solutions with the desired estimates. For any (epsilon >0), by constructing an example with (L^p) data but having no (L^{p+epsilon }) solutions, we verify the sharpness of the (L^p) regularity on the Hartogs triangle.

We prove that the area of each nonflat genus zero free boundary minimal surface embedded in the unit 3-ball is less than the area of its radial projection to ({mathbb {S}}^2). The inequality is asymptotically sharp, and we prove any sequence of surfaces saturating it converges weakly to ({mathbb {S}}^2), as currents and as varifolds.

In this paper, we investigate the weighted mass for weighted manifolds. By establishing a version of density theorem and generalizing Geroch conjecture in the setting of P-scalar curvature, we are able to prove the positive weighted mass theorem for weighted manifolds, which generalizes the result of Baldauf–Ozuch (Commun Math Phys 394(3):1153–1172, 2022) to non-spin manifolds.

The optimal control problems for the wave equation are considered on networks. The turnpike property is shown for the state equation, the adjoint state equation as well as the optimal cost. The shape and topology optimization is performed for the network with the shape functional given by the optimality system of the control problem. The set of admissible shapes for the network is compact in finite dimensions, thus the use of turnpike property is straightforward. The topology optimization is analysed for an example of nucleation of a small cycle at the internal node of network. The topological derivative of the cost is introduced and evaluated in the framework of domain decomposition technique. Numerical examples are provided.

The main aim of this paper is to investigate Hardy-Littlewood type Theorems and a Hopf type lemma on functions induced by a differential operator. We first prove more general Hardy-Littlewood type theorems for the Dirichlet solution of a differential operator which depends on (alpha in (-1,infty )) over the unit ball (mathbb {B}^n) of (mathbb {R}^n) with (nge 2), related to the Lipschitz type space defined by a majorant which satisfies some assumption. We find that the case (alpha in (0,infty )) is completely different from the case (alpha =0) due to Dyakonov (Adv. Math. 187 (2004), 146–172). Then a more general Hopf type lemma for the Dirichlet solution of a differential operator will also be established in the case (alpha >n-2).