The Journal of Geometric Analysis最新文献

Using a natural representation of a 1/s-concave function on ({mathbb {R}}^d) as a convex set in ({mathbb {R}}^{d+1},) we derive a simple formula for the integral of its s-polar. This leads to convexity properties of the integral of the s-polar function with respect to the center of polarity. In particular, we prove that the reciprocal of the integral of the polar function of a log-concave function is log-concave as a function of the center of polarity. Also, we define the Santaló regions for s-concave and log-concave functions and generalize the Santaló inequality for them in the case the origin is not the Santaló point.

In this paper we consider the 3D Euler equations and we first prove a criterion for energy conservation for weak solutions, where the velocity satisfies additional assumptions in fractional Sobolev spaces with respect to the space variables, balanced by proper integrability with respect to time. Next, we apply the criterion to study the energy conservation of solution of the Beltrami type, carefully applying properties of products in (fractional and possibly negative) Sobolev spaces and employing a suitable bootstrap argument.

In this paper, we investigate the holonomy group of n-dimensional projective Finsler metrics of constant curvature. We establish that in the spherically symmetric case, the holonomy group is maximal, and for a simply connected manifold it is isomorphic to ({mathcal {D}}i!f hspace{-3pt} f_o({mathbb {S}}^{n-1})), the connected component of the identity of the group of smooth diffeomorphism on the ({n-1})-dimensional sphere. In particular, the holonomy group of the n-dimensional standard Funk metric and the Bryant–Shen metrics are maximal and isomorphic to ({mathcal {D}}i!f hspace{-3pt} f_o({mathbb {S}}^{n-1})). These results are the firsts describing explicitly the holonomy group of n-dimensional Finsler manifolds in the non-Berwaldian (that is when the canonical connection is non-linear) case.

We establish a framework for fiberwise symmetrization to find a lower bound of a Dirichlet-type energy functional in a variational problem on a fibred Riemannian manifold, and use it to prove a comparison theorem of the first eigenvalue of the Laplacian on a warped product manifold.

We extend a systolic inequality of Guth for Riemannian manifolds of maximal ({mathbb {Z}}_2) cup-length to piecewise Riemannian complexes of dimension 2. As a consequence we improve the previous best universal lower bound for the systolic area of groups for a large class of groups, including free abelian and surface groups, most of irreducible 3-manifold groups, non-free Artin groups and Coxeter groups or, more generally, groups containing an element of order 2.

We consider the free boundary problem for a layer of incompressible fluid lying below the atmosphere and above a rigid bottom in the horizontally infinite setting. The fluid dynamics is governed by the incompressible Euler equations with damping and gravity, and the effect of surface tension is neglected on the upper free boundary. We prove the global well-posedness of the problem with the small initial data in both 2D and 3D. One of key ideas here is to make use of the time-weighted dissipation estimates to close the nonlinear energy estimates; in particular, this implies that the Lipschitz norm of the velocity is integrable-in-time, which is significantly different from that of viscous surface waves (Guo and Tice in Anal PDE 6(6):1429–1533, 2013; Wang in Adv Math 374:107330, 2020).

By refining the volume estimate of Heintze and Karcher [11], we obtain a sharp pinching estimate for the genus of a surface in (mathbb S^{3}), which involves an integral of the norm of its traceless second fundamental form. More specifically, we show that if g is the genus of a closed orientable surface (Sigma ) in a 3-dimensional orientable Riemannian manifold M whose sectional curvature is bounded below by 1, then (4 pi ^{2} g(Sigma ) le 2left( 2 pi ^{2}-|M|right) +int _{Sigma } f(|{mathop {A}limits ^{circ }}|)), where ( {mathop {A}limits ^{circ }} ) is the traceless second fundamental form and f is an explicit function. As a result, the space of closed orientable embedded minimal surfaces (Sigma ) with uniformly bounded (Vert AVert _{L^3(Sigma )}) is compact in the (C^k) topology for any (kge 2).

We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in ({mathbb {R}}^{3}) can be extended to an infinitely narrow flat ribbon having minimal bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish.

In this paper, we are concerned with the energy equality for axisymmetric weak solutions of the 3D Navier–Stokes equations. The classical Shinbrot condition says that if the weak solution u of the Navier–Stokes equations belongs (L^q(0,T;L^p(mathbb {R}^3))) with (frac{1}{q}+frac{1}{p}=frac{1}{2}) and (pge 4), then u must satisfy the energy equality. For the axisymmetric Navier–Stokes equations, in our previous paper, we found that it is enough to impose the Shinbrot condition to (tilde{u}=u^re_r+u^z e_z). The recent papers (Chiun-Chuan et al., Commun PDE 34(1–3):203–232, 2009; Koch et al., Acta Math 203(1):83–105, 2009) tell us if

then u is smooth , therefore the energy equality holds. It is natural to ask the relation between a priori bound on the velocity and the energy conservation. The aim of this paper is to investigate this problem. We shall prove that if

and

then the energy equality holds.

A standard technique for producing monogenic functions is to apply the adjoint quaternionic Fueter operator to harmonic functions. We will show that this technique does not give a complete system in (L^2) of a solid torus, where toroidal harmonics appear in a natural way. One reason is that this index-increasing operator fails to produce monogenic functions with zero index. Another reason is that the non-trivial topology of the torus requires taking into account a cohomology coefficient associated with monogenic functions, apparently not previously identified because it vanishes for simply connected domains. In this paper, we build a reverse-Appell basis of harmonic functions on the torus expressed in terms of classical toroidal harmonics. This means that the partial derivative of any element of the basis with respect to the axial variable is a constant multiple of another basis element with subindex increased by one. This special basis is used to construct respective bases in the real (L^2)-Hilbert spaces of reduced quaternion and quaternion-valued monogenic functions on toroidal domains.