In this paper we extend a recent result of Collin-Rosenberg ({\it a solution to the minimal surface equation in the Euclidean disc has radial limits almost everywhere}) to a large class of differential operators in Divergence form. Moreover, we construct an example (in the spirit of \cite{CR2}) of a minimal graph in $\mr$, where $\m$ is a Hadamard surface, over a geodesic disc which has finite radial limits in a mesure zero set.
In this paper we study a class of functions that appear naturally in some equidistribution problems and that we call $F$-harmonic. These are functions of the universal cover of a closed and negatively curved which possess an integral representation analogous to the Poisson representation of harmonic functions, where the role of the Poisson kernel is played by a Hölder continuous kernel. More precisely we prove a theorem à la Fatou about the nontangential convergence of quotients of such functions, from which we deduce some basic properties such as the uniqueness of the $F$-harmonic function on…
We prove that there are no minimal hypersurfaces properly immersed in any region of the Euclidean space bounded by unstable minimal cones. We also prove the analogous result for $r$-minimal hypersurfaces.
In this paper we investigate surfaces in $\mathbb C P^2$ without complex points and characterize the minimal surfaces without complex points and the minimal Lagrangian surfaces by Ruh-Vilms type theorems. We also discuss the liftability of an immersion from a surface to $\mathbb C P^2$ into $S^5$ in Appendix A.
In this paper, we consider a Generalized Bernstein Theorem for a type of generalized minimal surfaces, namely minimal Plateau surfaces. We show that if an orientable minimal Plateau surface is stable and has quadratic area growth in $\mathbb{R}^3 $, then it is flat.
In this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a Riemannian three-manifold. This removable singularity theorem is the key result used in our proof that a complete, embedded minimal surface in $\mathbb{R}^3$ with quadratic decay of curvature has finite total curvature.
We apply the local removable singularity theorem for minimal laminations and the local picture theorem on the scale of topology to obtain two descriptive results for certain possibly singular minimal laminations of $\mathbb{R}^3$. These two global structure theorems will be applied in forthcoming papers to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in $\mathbb{R}^3$, and to prove that a complete, embedded minimal surface in $\mathbb{R}^3$ with finite genus and a countable number of ends is proper.
We provide a congruence theorem for minimal surfaces in $S^5$ with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations for minimal surfaces in $S^5$ with constant contact angle satisfy an equation for the Laplacian of the holomorphic angle. Also, we will give a characterization of flat minimal surfaces in $S^5$ with constant contact angle.
Built on a recent work of Almaraz, Barbosa, de Lima on positive mass theorems on asymptotically flat manifods with a noncompact boundary, we apply free boundary minimal surface techniques to prove their positive mass theorem and study the existence of positive scalar curvature metrics with mean convex boundary on a connected sum of the form $(\mathbb{T}^{n-1}\times[0,1])\#M_0$.
In this paper we have proved several approximation theorems for the family of minimal surfaces in R^3 that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of C^k convergence on compact sets, for any k. As a consequence of the above density result, we have been able to produce the first example of a complete proper minimal surface in R^3 with uncountably many ends.
Using Schauder's theory for linear elliptic partial differential equations in two independent variables and fundamental estimates for univalent mappings due to E. Heinz we establish an upper bound of the Gaussian curvature of two-dimensional minimal surface graphs in R^n. This leads us to a theorem of Bernstein-Liouville type.
We show that, among free boundary minimal surfaces in the unit ball in the three-dimensional Euclidean space, the flat equatorial disk and the critical catenoid are characterised by a pinching condition on the length of their second fundamental form.
We prove that if $M$ is a strictly stable complete minimal hypersurface in Euclidean space with finite density at infinity and which lies on one side of a minimal cylinder with cross-section a strictly stable area minimizing hypercone, then $M$ must be cylindrical. Applications will be given in the references [Sim20a], [Sim20b].
The purpose of this paper is to study a complete orientable minimal hypersurface with finite index in an $(n+1)$-dimensional Riemannian manifold $N$. We generalize Theorems 1.5-1.6 (\cite{Seo14}). In 1976, Schoen and Yau proved the Liouville type theorem on stable minimal hypersurface, i.e., Theorem 1.7 (\cite{SchoenYau1976}). Recently, Seo (\cite{Seo14}) generalized Theorem 1.7 (\cite{SchoenYau1976}). Finally, we generalize Theorems 1.7 (\cite{SchoenYau1976}) and 1.8 (\cite{Seo14})
Given two univalent harmonic mappings $f_1$ and $f_2$ on $\mathbb{D}$, which lift to minimal surfaces via the Weierstrass-Enneper representation theorem, we give necessary and sufficient conditions for $f_3=(1-s)f_1+sf_2$ to lift to a minimal surface for $s\in[0,1]$. We then construct such mappings from Enneper's surface to Scherk's singularly periodic surface, Sckerk's doubly periodic surface to the catenoid, and the 4-Enneper surface to the 4-noid.
We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil_3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil_3 is entire. Also, the sister surface of an entire minimal graph in Nil_3 is an entire constant mean curvature 1/2 graph in H^2 x R, and conversely. This gives a classification of all entire constant mean curvature 1/2 graphs in H^2 x R. Finally we construct properly embedded constant mean curvature 1/2 …
We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil_3 and the Lie group Sol_3 endowed with their left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil_3 that lies on one side of some entire minimal graph G, then S is the image of G by a vertical translation. If S is a properly immersed minimal surface in Sol_3 that lies on one side of a special plane, then S is another special plane.
Let $M$ be an $n$-dimensional smooth oriented complete embedded minimal hypersurface in $\mathbb{R}^{n+1}$ with Euclidean volume growth. We show that if the image under the Gauss map of $M$ avoids some neighborhood of a half-equator, then $M$ must be an affine hyperplane.
Combining the tools of geometric analysis with properties of Jordan angles and angle space distributions, we derive a spherical and a Euclidean Bernstein theorem for minimal submanifolds of arbitrary dimension and codimension, under the condition that the Gauss image is contained in some geometrically defined closed region of a Grassmannian manifold. The proof depends on the subharmoncity of an auxiliary function, the Codazzi equations and geometric measure theory.
In this paper we prove two theorems. The first one is a structure result that describes the extrinsic geometry of an embedded surface with constant mean curvature (possibly zero) in a homogeneously regular Riemannian three-manifold, in any small neighborhood of a point of large almost-maximal curvature. We next apply this theorem and the Quadratic Curvature Decay Theorem (previously proven by the same authors in [14]) to deduce compactness, descriptive and dynamics-type results concerning the space $D(M)$ of non-flat limits under dilations of any given properly embedded minimal surface $M$ in …
In this paper, we study complete oriented $f$-minimal hypersurfaces properly immersed in a cylinder shrinking soliton $(\mathbb{S}^n\times \mathbb{R}, \bar{g}, f)$. We prove that such hypersurface with $L_f$-index one must be either $\mathbb{S}^n\times\{0\}$ or $\mathbb{S}^{n-1}\times\mathbb{R}$, where $\mathbb{S}^{n-1}$ denotes the sphere in $\mathbb{S}^{n}$ of the same radius. Also we prove a pinching theorem for them.
In this note, we study minimal Lagrangian surfaces in $\mathbb{B}^4$ with Legendrian capillary boundary on $\mathbb{S}^3$. On the one hand, we prove that any minimal Lagrangian surface in $\mathbb{B}^4$ with Legendrian free boundary on $\mathbb{S}^3$ must be an equatorial plane disk. One the other hand, we show that any annulus type minimal Lagrangian surface in $\mathbb{B}^4$ with Legendrian capillary boundary on $\mathbb{S}^3$ must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng (Sci. China Math., 2020).
In this note, we investigate the well-known Yau rigidity theorem for minimal submanifolds in spheres. Using the parameter method of Yau and the DDVV inequality verified by Lu, Ge and Tang, we prove that if $M$ is an $n$-dimensional oriented compact minimal submanifold in the unit sphere $S^{n+p}(1)$, and if $K_{M}\geq\frac{sgn(p-1)p}{2(p+1)},$ then $M$ is either a totally geodesic sphere, the standard immersion of the product of two spheres, or the Veronese surface in $S^4(1)$. Here $sgn(\cdot)$ is the standard sign function. We also extend the rigidity theorem above to the case where $M$ is a…
We prove that in a Riemannian manifold $M$, each function whose Hessian is proportional the metric tensor yields a weighted monotonicity theorem. Such function appears in the Euclidean space, the round sphere $S^n$ and the hyperbolic space $\mathbb{H}^n$ as the distance function, the Euclidean coordinates of $\mathbb{R}^{n+1}$ and the Minkowskian coordinates of $\mathbb{R}^{n,1}$. Then we show that weighted monotonicity theorems can be compared and that in the hyperbolic case, this comparison implies three $SO(n,1)$-distinct unweighted monotonicity theorems. From these, we obtain upper bounds …
In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere.
We consider minimal maps $f:M\to N$ between Riemannian manifolds $(M,\mathrm{g}_M)$ and $(N,\mathrm{g}_N)$, where $M$ is compact and where the sectional curvatures satisfy $\sec_N\le σ\le \sec_M$ for some $σ>0$. Under certain assumptions on the differential of the map and the second fundamental form of the graph $Γ(f)$ of $f$, we show that $f$ is either the constant map or a totally geodesic isometric immersion.
Let M be a closed minimal hypersurface in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature. We prove that, if the sum of the cubes of all principal curvatures and the number of distinct principal curvatures are constant, then M is isoparametric. Moreover, We give all possible values for squared length of the second fundamental form of M. This result provides another piece of supporting evidence to the Chern conjecture.
For $n\geq2,$ we obtain Liouville type theorems for minimal surface equations in half space $\mathbf R^n_+$ with affine Dirichlet boundary value or constant Neumann boundary value.
In this paper, we prove a uniform approximation theorem with interpolation for complete conformal minimal surfaces with finite total curvature in the Euclidean space $\mathbb{R}^n$ $(n\ge 3)$. As application, we obtain a Mittag-Leffler type theorem for complete conformal minimal immersions $M\to\mathbb{R}^n$ on any open Riemann surface $M$.
Using an analogue of Myers' theorem for minimal surfaces and three dimensional topology, we prove the diameter sphere theorem for Ricci curvature in dimension three and a corresponding eigenvalue pinching theorem. This settles these two open problems for closed 3 manifolds with positive Ricci curvature since they are both false in dimensions greater than three.
The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded.
We prove the two theorems of the title, settling two long standing questions in the local theory of singular minimal hypersurfaces. The sharpness of either result is with respect to its hypothesis on the size of the allowable singular sets. The proofs of both theorems rely heavily on the author's recent regularity and compactness theory for stable minimal hypersurfaces, and on earlier work of Ilmanen, Simon and Solomon--White.
Let $Σ$ be a complete Riemannian manifold with the volume doubling property and the uniform Neumann-Poincar$\mathrm{\acute{e}}$ inequality. We show that any positive minimal graphic function on $Σ$ is a constant.
The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.
In this paper, we prove some Bernstein type results for $n$-dimensional minimal Lagrangian graphs in quaternion Euclidean space $H^n\cong R^{4n}$. In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in $C^n$
We extend the well-known Denjoy-Ahlfors theorem on the number of different asymptotic tracts of holomorphic functions to subharmonic functions on arbitrary Riemannian manifolds. We obtain some new versions of the Liouville theorem for $\p$-harmonic functions without requiring the geodesic completeness requirement of a manifold. Moreover, an upper estimate of the topological index of the height function on a minimal surface in $\R{n}$ has been established and, as a consequence, a new proof of Bernstein's theorem on entire solutions has been derived. Other applications to minimal surfaces are al…
Given any $n \geq 2$, we show that if $Ω\subsetneq \mathbb{R}^n$ is an open convex domain (e.g. a half-space), and $u : Ω\to \mathbb{R}$ is a solution to the minimal surface equation which agrees with a linear function on $\partial Ω$, then $u$ must itself be linear.
In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold $γ$ contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound $γ$. In order to do so, we develop min-max methods similar to those of De Lellis and Ramic, references in the paper, adapted to the discrete setting of Almgren and Pitts.
In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space $\mathbb{E}^3$ is the graph of a function over a convex domain, then each surface of its associated family is also a graph. The same is true for maximal surfaces in the Minkowski 3-space $\mathbb{L}^3$. In this article, we introduce a new deformation family that continuously connects minimal surfaces in $\mathbb{E}^3$ and maximal surfaces in $\mathbb{L}^3$, and prove a Krust-type theorem for this deformation family. This result induces Krust-type theorems for various important deformatio…
In this paper we prove that a complete minimal surface immersed in H^2xR, with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a horizontal catenoid. Moreover, we give a geometric description of minimal ends of finite total curvature in H^2xR. We also prove that a minimal complete end E with finite total curvature is properly immersed and that the Gaussian curvature of E is locally bounded in terms of the geodesic distance to its boundary.
We prove a Bernstein-type theorem for two-valued minimal graphs in the four-dimensional Euclidean space $\mathbf{R}^4$. This states that two-valued functions defined on the entire $\mathbf{R}^3$, and whose graph is a minimal surface, must necessarily be linear. This is a two-valued analogue of the classical Bernstein theorem, which asserts that in dimensions up to $n+1 \leq 8$, an entire single-valued minimal graph is linear. The main contrast with the single-valued theory is the presence of a large set of singularities in the graphs of two-valued functions. Indeed two-valued minimal graphs ar…
The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Koll'ar's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.
In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space-time transformations. In this second part, we shall adapt the formalism developed in the first paper to the context of gauge theories and deal with minimal coupling and Utiyama's theorem.
