Regular Riemannian Manifolds Research Articles

Rectifiability of Flat Singular Points for Area-Minimizing mod 2Q Hypercurrents

Abstract Consider an $ m $-dimensional area minimizing mod$ (2Q) $ current $ T $, with $ Q\in {\mathbb {N}} $, inside a sufficiently regular Riemannian manifold of dimension $ m + 1 $. We show that the set of singular density-$ Q $ points with a flat tangent cone is $ (m-2) $-rectifiable. This complements the thorough structural analysis of the singularities of area-minimizing hypersurfaces modulo $ p $ that has been completed in the series of works of De Lellis–Hirsch–Marchese–Stuvard and De Lellis–Hirsch–Marchese–Stuvard–Spolaor, and the work of Minter–Wickramasekera.

An upper Minkowski dimension estimate for the interior singular set of area minimizing currents

AbstractWe show that for an area minimizing m‐dimensional integral current T of codimension at least two inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most . This provides a strengthening of the existing ‐dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by‐product of the proof, we establish an improvement on the persistence of singularities along the sequence of center manifolds taken to approximate T along blow‐up scales.

A revisit to “On BMO and Carleson measures on Riemannian manifolds”

Let $\mathcal {M}$ be an Ahlfors $n$-regular Riemannian manifold such that either the Ricci curvature is non-negative or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. In the paper [IMRN, 2022, no. 2, 1245-1269] of Brazke–Schikorra–Sire, the authors characterised the BMO function $u : \mathcal {M} \to \mathbb {R}$ by a Carleson measure condition of its $\sigma$-harmonic extension $U:\mathcal {M}\times \mathbb {R}_+ \to \mathbb {R}$. This paper is concerned with the similar problem under a more general Dirichlet metric measure space setting, and the limiting behaviours of BMO & Carleson measure, where the heat kernel admits only the so-called diagonal upper estimate. More significantly, without the Ricci curvature condition, we relax the Ahlfors regularity to a doubling property, and remove the pointwise bound on the gradient of the heat kernel. Some similar results for the Lipschitz function are also given, and two open problems related to our main result are considered.

Cauchy problems for parabolic equations in Sobolev–Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds

In this paper we establish optimal solvability results, that is, maximal regularity theorems, for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless complete Riemannian manifolds with bounded geometry. We employ an anisotropic extension of the Fourier multiplier theorem for arbitrary Besov spaces introduced in earlier by the author. This allows for a unified treatment of Sobolev-Slobodeckii and little H\"older spaces. In the flat case we recover classical results for Petrowskii-parabolic Cauchy problems.

Some remarks on uniformly regular Riemannian manifolds

We establish the equivalence between the family of uniformly regular Riemannian manifolds without boundary and the class of manifolds with bounded geometry.

Minimal immersions of compact bordered Riemann surfaces with free boundary

Let N N be a complete, homogeneously regular Riemannian manifold of dim N ≥ 3 N \geq 3 and let M M be a compact submanifold of N N . Let Σ \Sigma be a compact orientable surface with boundary. We show that for any continuous f : ( Σ , ∂ Σ ) → ( N , M ) f: \left ( \Sigma , \partial \Sigma \right ) \rightarrow \left ( N, M \right ) for which the induced homomorphism f ∗ f_{*} on certain fundamental groups is injective, there exists a branched minimal immersion of Σ \Sigma solving the free boundary problem ( Σ , ∂ Σ ) → ( N , M ) \left ( \Sigma , \partial \Sigma \right ) \rightarrow \left ( N, M \right ) , and minimizing area among all maps which induce the same action on the fundamental groups as f f . Furthermore, under certain nonnegativity assumptions on the curvature of a 3 3 -manifold N N and convexity assumptions on the boundary M = ∂ N M=\partial N , we derive bounds on the genus, number of boundary components and area of any compact two-sided minimal surface solving the free boundary problem with low index.

Continuous maximal regularity on uniformly regular Riemannian manifolds

We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on uniformly regular Riemannian manifolds M. As an application, we show that solutions to the Yamabe flow on M instantaneously regularize and become real analytic in space and time. The regularity result is obtained by introducing a family of parameter-dependent diffeomorphisms acting on functions on M in conjunction with maximal regularity and the implicit function theorem.

Some sufficient conditions for immersion in the large of metrics of positive curvature

It is proved that a regular Riemannian manifold diffeomorphic to a circle and having positive Gaussian curvature bounded from zero is immersible into a three-dimensional Euclidean space in the form of a regular surface if it has smallLp (the norm of the gradient of Gaussian curvature), p > 2, or if it has a sufficiently small area (with any behavior of the geodesic boundary curvature).

The existence of embedded minimal surfaces and the problem of uniqueness

It was in the early nineteen thirties that Douglas and Radd solved the Plateau problem for a rectifiable Jordan curve in Euclidean space. The minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk. In 1948 Morrey generalized the theorem of Douglas and Radd to minimal surfaces in a homogeneously regular Riemannian manifold. In both cases the solutions are defective as they may possess branch points. It was not until 1968 that Osserman [-24] was able to prove that interior branch points do not exist on the Douglas-Radd-Morrey solution. (Osserman's theorem in this full generality was proved by Alt [-2, 3] and Gulliver [8].) If the Jordan curve is real analytic, Gulliver and Lesley [9] showed that the solution surface is free of branch points even on its boundary. Hence, at least in this case, we know that the Douglas-Radd-Morrey solution is an immersion and represents a classical surface. However, the Douglas-Radd-Morrey solution may in general have selfintersections. It was then a classical problem to find conditions on the Jordan curve to guarantee that the solution is an embedded disk. In 1932 Radd was the first to produce a general condition of this type. He proved that if the curve can be projected onto a convex curve in the plane, then it bounds a unique embedded minimal disk which is a graph over this plane. It was not until the seventies that there was further progress on the problem of embeddedness. Osserman had conjectured that if a Jordan curve is extreme, that is, lies on the boundary of its convex hull, then the Douglas-Rad6 solution is embedded. Gulliver and Spruck [10] proved this conjecture under the additional assumption that the absolute total curvature of the Jordan curve is less than 4~z. Right after this work, Tomi and Tromba [32] showed that an extreme curve must bound an embedded minimal disk which need not be stable. In 1976 Almgren and Simon [1] established the existence for such a curve of an embedded stable minimal disk (which need not be a Douglas-Radd solution). Finally in the same year, the authors [18] proved that any Douglas-Rad6 solution must be embedded.

THE EXTERIOR DIAMETER OF AN IMMERSED RIEMANNIAN MANIFOLD

In this paper we establish a general lower bound for the radius of the sphere containing a complete regular Riemannian manifold in Euclidean space n$ SRC=http://ej.iop.org/images/0025-5734/21/3/A07/tex_sm_2027_img2.gif/>.Bibliography: 10 titles.

Theory of Bessel potentials. III: Potentials on regular manifolds

Abstract : The present part of the 'Theory of Bessel Potentials' contains Chapter 4 dealing with potentials on regular Riemannian manifolds. (Author)

The Problem of Plateau on a Riemannian Manifold

The problem of Plateau or the proof of the existence of a surface of least area bounded by a given Jordan contour in Euclidean space is one of the classical problems of the Calculus of Variations. This problem has been studied at least since the time of Riemann but it was not until 1930 that satisfactory proofs for a general contour were given independently by Douglas [25]' and Rado [15]. However, before that time, considerable information was gathered and the problem was solved in many interesting special cases. For the literature of this period, we refer to the excellent report of Rad6 [38] on the problem of Plateau. Shortly after the appearance of the solutions of Douglas and Rado, McShane presented an interesting solution which was perhaps more along the lines of the straightforward direct methods of the Calculus of Variations (see [30]). In the past seventeen years the problem has been generalized by replacing r by several contours r1, * * *, rk and by prescribing the topological structure of the desired surface ([5], [19], [27], [28], 129], [33], 135], [41], [43]), or by replacing the condition that the surface be bounded by a given contour by the condition that only parts of the boundary be prescribed, the remainder of the boundary being merely restricted to lie along certain manifolds (see [18], [22], [23], [24]). Finally minimal surfaces which do not furnish even relative minima have been discussed (see [20], [21], [31], [32], [33], [34], [35], [39], [40], [42], [43]). The present paper generalizes the problem by replacing the underlying Euclidean space by a Riemannian manifold of considerable generality. The writer knows of only one paper in which the Plateau problem has been solved in any space other than Euclidean space, namely the recent paper by A. Lonseth [7] where the space considered is hyperbolic space. However, S. Bochner [2] has studied the harmonic functions in a general Riemannian metric (of sufficient differentiability) but has not demonstrated their minimizing property in general. This work is, of course of great interest as it is still true in a Riemannian space (of sufficient differentiability) that if a minimal surface is mapped conformally on a plane region, then the representing vector is harmonic in Bochner's sense and also that a surface of least area (if of sufficient differentiability) is a minimal surface (in the sense of Differential Geometry). The present writer does not attempt to generalize all of the preceding results on the Plateau problem but restricts himself to the case of a surface of the type of a k-fold connected plane region bounded by k given contours. The space is a homogeneously (see Def. 3.1) Riemannian manifold 9) of class C'. Every compact, regular Riemannian manifold of class C' is seen to be homo-