This paper shows that any m-dimensional rectifiable flat chain modulo four in R^{m+1} that minimizes an even parametric elliptic functional F is, except for a closed singular set of m-2 dimensional measure 0, a smoothly immersed manifold whose sheets are locally F minimizing. The proof uses a decomposition lemma, according to which the mod 4 variational problem is locally equivalent to two mod 2 variational problems.
The following decomposition theorem is proved: in a riemannian manifold, a codimension one rectifiable cycle that is a boundary modulo p can be decomposed into p homologous rectifiable cycles. As corollaries one has: (1) the aforesaid boundary regularity (which extends the work of Hardt and Simon) and (2) any hypersurface that minimizes a parametric elliptic functional among integral currents also minimizes among real flat chains.
The title assertion is deduced (using an idea of Reifenberg) from an ``epiperimetric'' inequality. The epiperimetric inequality is proved by constructing a comparison surface from the graph of a multivalued harmonic function, the area of which is estimated in terms of the Fourier series of its boundary values.
This paper shows that every smooth embedding f of the boundary of B^{m} into R^{m+1} (with m greater than 2 and less than 7 extends to a lipschitz map g from B^{m} to R^{m+1} that minimizes m-dimensional mapping area among all such maps. Indeed, if T is an area-minimizing integral current with the given boundary, then there is a lipschitz extension g of f whose image consists of T together with an (m-1)-dimensional set S. Since S does not contribute to the m-dimensional area, the least area among surfaces of arbitrary topological type is actually attained by a lipschitz image of B^{m}.
If G is a closed curve in R^{m}, let a(G)
denote the least area (counting multiplicities) of any oriented surface
with boundary G. This paper contains, for each n>1, an
example of a smooth simple closed curve G in R^{4} such
that
Let M be a compact connected oriented manifold with (possibly empty) boundary and with dimension at least 3. Let X be a simply connected riemannian manifold, and let f be a lipschitz map from M to X. Let F be the set of maps g that are homotopic to f under lipschitz homotopies H:[0,1] x M --> X such that H(t,x)=f(x) for x in the boundary of M. This paper shows that the infimum mapping area of maps g in F is equal to the infimum area of m-dimensional integral currents T such that T-f_{#}([M]) = boundary(R) for some (m+1)-dimensional integral current R in X. Furthermore, if the restriction of f to boundary(M) is one-to-one and if the infimum is attained by a sufficiently regular current T, then it is attained by a map g whose image is T together with a singular set of dimension at most (m-1). Similar results are obtained for non-orientable M and non-simply connected X.
This paper shows that for almost every smooth closed curve in R^{n} (or in a riemannian manifold), the unoriented area minimizing surfaces (i. e., flat chains mod 2) that it bounds have no singularities.
Research announcement of results such as: an n-dimensional soap-film (i. e., (M,e,d)-minimizing set) in an (n+1)-dimensional manifold consists of: (1) smooth hypersurfaces, (2) smooth (n-1)- manifolds along which 3 of the hypersurfaces meet at equal angles, (3) smooth (n-2)-manifolds along which 6 of the hypersurfaces and 4 of the (n-1)-manifolds meet at equal angles, and (4) a set (possibly empty) of dimension at most n-3.
Research announcement. See items 10 and 15.
Let X(f) be the L^{p}-norm of |Df| (or a similar functional of f), and let M and N be compact riemannian manifolds. This paper shows that the infimum of X(g) among all lipschitz maps g: M --> N homotopic to a given map f does not depend on the homotopy class of f; it only depends on the [p]-homotopy class of f, i. e. on the homotopy class of the restriction of f to the [p]-dimensional skeleton of (some triangulation of) M. (Here [p] is the greatest integer less than or equal to p.) Indeed, the infimum is 0 if and only if f is [p]-homotopic to a constant map. It follows that the identity map on M is homotopic to maps f with X(f) arbitrarily small if and only if the kth homotopy group of M vanishes for k=1,2,...,[p].
Let T be an m-dimensional hypersurface that minimizes an even parametric elliptic functional modulo p. This paper shows that if T is, in some cylinder, weakly near a disk with multiplicity k < p/2, then T is regular in a smaller cylinder. For odd p it follows that the dimension of the singular set is < m - e (for some e > 0 independent of the surface) and is at most m-1 in case the functional is m-dimensional area.
Let M be a compact connected manifold with nonempty boundary, N be a riemannian manifold, and let S be the set of equivalence classes [f] of C^{j,a} mappings f:M--> N that are minimal (i. e. stationary) with respect to area (or some other parametric elliptic functional), where two maps are equivalent if they coincide on bdry M and if they both parametrize the same surface in N. This paper shows that S has the structure of a Banach manifold, and uses that structure to obtain certain genericity results: (1) it is exceptional for a curve in R^{n} to bound any unoriented area-minimizing surfaces (flat chains mod 2) with singularities, and (2) it is exceptional for a boundary to bound two minimal immersed surfaces with the same area, or any minimal immersed surfaces with nonzero normal Jacobi fields that vanish on bdry M.
Let M be a complete oriented two dimensional riemannian manifold. Let K denote the scalar curvature and, in case M is an immersed submanifold of R^{n}, let B denote the second fundamental form. This paper gives simple proofs that: (1) if the integral of |K| is finite, then M is of finite topological type (i.e., M is homeomorphic to a compact surface N minus finitely many points); (2) if the integral of |B|^{2} is finite, then the integral of K is a multiple of 2\pi (or 4\pi in case n=3); (3) if the integral of |B|^{2} is finite and M is non-positively curved with respect to each normal direction, then M is properly immersed and the gauss map extends continuously to all of N. An example shows that in case (2), the gauss map need not extend. Theorem 1 was originally proved by Huber. For minimal surfaces M, theorems 2 and 3 were discovered by Chern and Osserman.
This paper gives a fairly elementary proof that if F is an even parametric elliptic functional and if M is an embedded F- stationary surface in a 3-manifold N with brdy M contained in brdy N, then the principal curvatures of M are bounded in terms of F, N, the area and the genus of M, and brdy M. This implies a compactness theorem for such surfaces.
Let M be a compact riemannian manifold and N be a compact
submanifold of a euclidean space E.
Let L^{1, p}(M,N) be the
space of maps f:M--> E such that f(x)\in N for every
x\in M and such that
Similar results are proved for the weak-bounded closure H^{1, p}(M,N) of Lip(M,N) in L^{1, p}(M,N).
Let F be an even parametric elliptic functional. We show that if M and N are F-stationary hypersurfaces that touch each other without crossing, and if one of them is regular, then then they coincide locally.
Among other results, this paper shows that for most of the known examples of area minimizing hypercones C with isolated singularities, there exists a complete singular minimal hypersurface that is asymptotic to C at infinity but that is not a cone, and a complete area minimizing hypersurface that is asymptotic to C at infinity but that is neither a cone nor any leaf of the Hardt-Simon minimal foliation associated to a cone.
This paper gives a new proof of the compactness theorem that, unlike previous proofs, uses neither the structure theorem for sets of finite hausdorff measure nor the theory of multi\-valued functions. (See item 44 for a more recent elementary proof that also applies to finite coefficient groups.)
This paper proves the title assertion.
Let f be a harmonic map from the three dimensional unit ball to a compact smooth two dimensional manifold N, such that f has an isolated singularity at 0. This paper shows that as e --> 0. the maps f_{e}(x)=f(e x) for |x|=1 converge rapidly (i. e., like e^{a} for some a > 0) to a harmonic map from the two-sphere to N. An example is given of an analytic three dimensional manifold N and a harmonic map f for which the convergence is slower than any power of |x|. (L. Simon proves convergence in all dimensions provided N is analytic.)
This paper describes the evolution of curves and surfaces by mean curvature, Almgren's proof of the optimal isoperimetric inequality, and new examples (Kapouleas, Wente) of constant mean curvature surfaces.
Let f be a proper isometric minimal embedding in R^{3} of a complete surface M with more than one end. This paper shows that if g is any isometric minimal immersion of M in R^{3}, then g = R f for some ambient isometry R: R^{3} --> R^{3}.
Let G be a pair of convex curves in parallel planes in R^{3}. This paper shows that G either (i) bounds no minimal annuli, (ii) bounds exactly one minimal annulus, which is barely stable, or (iii) bounds exactly two minimal annuli, one strictly stable and the other of index 1 and nullity 0. It is not known whether such a G can bound non-annular connected minimal surfaces, but we show that if G has a plane of reflective symmetry, then any stable minimal surface that it bounds must be an annulus.
This paper proves that if N is a compact 3-manifold with boundary, F is an even parametric elliptic functional on N such that bdry(N) is F-convex, and if S is a smooth embedded curve in bdry(N) that bounds embedded disks in N, then S bounds a smooth embedded disk D that minimizes F(D). A similar result is proved for higher genus surfaces.
Let S be the space of ordered pairs
This paper proves results such as: if a smooth embedded two-sphere in the boundary of the 4-ball bounds a unique smooth stable minimal hypersurface, then that hypersurface is a homology ball.
We prove, for instance, that an n-dimensional soap-film (i. e., (M,e,d)-minimizing set) in an (n+1)-dimensional manifold consists of: (1) smooth hypersurfaces, (2) smooth (n-1)- manifolds along which 3 of the hypersurfaces meet at equal angles, (3) smooth (n-2)-manifolds along which 6 of the hypersurfaces and 4 of the (n-1)-manifolds meet at equal angles, and (4) a set (possibly empty) of dimension at most (n-3).
This paper gives an example of a harmonic map f from the 3-ball to a smooth 4 manifold such that f has a continuum of distinct tangent maps at an isolated singularity. No examples of such nonuniqueness were known previously.
Let G be a pair of convex plane curves in R^{3} such that G lies on the boundary of its convex hull. We show that G bounds either (i) no minimal annuli, or (ii) exactly one minimal annulus, which has nullity 1 and index 0, or (iii) exactly two minimal annuli, one stricly stable and the other of index 1 and nullity 0. We also give an example of a pair G of simple closed curves in parallel planes, each curve having total curvature less than 4\pi + \epsilon, such that G bounds a stable nonembedded minimal annulus. We also prove that if S is the space of embedded minimal annuli with boundary curves in a (fixed) pair of parallel planes, then S is contractible.
This paper gives an example of a smooth 5-manifold N such that for an open set of maps f from the 3-sphere to N, f extends to one or more energy minimizing maps of the 4-ball into N, and each such map has a continuum of distinct tangent maps at an isolated singularity.
If a minimal surface in
R^{2n+2}=C^{n+1} has a branch point of order
Q-1 at the origin, then by rotating the coordinate axes the surface may
be written as {(z^{Q},f(z)): z in U } near 0,
where f:U --> C^{n} vanishes to order > or = Q+1.
This paper shows that f is
either a function of z^{Q} or else has the form
The techniques developed are also used to prove that every pseudoholomorphic curve in an almost complex 4-manifold must be locally equivalent (via a C^{1} ambient diffeomorphism) to an algebraic curve. This strengthens somewhat a result of D. McDuff.
Note: For readers interested only in the pseudoholomorphic curves, the proof can be streamlined. See the preprint "Singularities of J-holomorphic curves" by J. C. Sikorav. Sikorav also gives examples showing that C^{1} cannot be improved to C^{2}.
If N is a riemannian manifold and M is smooth compact submanifold (with or without boundary) that is stationary and strictly stable for the area functional, we show that there is an open subset U of N containing M such that M has less area than any other surface in U that is homologous (in U) to M. Similarly, if M is unstable but has nullity 0, then it is the unique solution of a certain minimax problem in an open subset U of N. The theorems also hold when area is replaced by other parametric elliptic functionals.
We show that it is possible to connect strictly stable minimal surfaces along prescribed bridges provided (roughly speaking) the bridges are sufficiently thin and meet the surface at angles strictly between 0 and 2\pi. The proof works in all dimensions and codimensions and also for surfaces that are minimal or have constant mean curvature with respect to riemannian metrics, or, more generally, that are stationary for parametric elliptic functionals.
The results of the previous paper are extended to allow unstable surfaces of nullity 0. We use the theorem to show that there is a simple closed curve in bdry(B^{3}) that is smooth except at one point and a finite number c with the following property. For every g, i, and A > c, the curve bounds uncountably many embedded minimal surfaces of genus g, index i, and area A. We extend the bridge theorem to allow possibly singular surfaces, provided they are uniquely minimizing (as currents) in some open subset of the ambient space.
An announcement of results such as: the singular set in spacetime of an n-dimensional initially mean-convex hypersurface flowing by mean curvature has hausdorff dimension at most n-1.
Certain triply periodic surfaces are shown to flow smoothly to triply periodic minimal surfaces, whereas others disappear in finite time.
This paper analyses the way in which the topology of hypersurfaces change as a result of singularities in flows such as the mean curvature flow.
This paper shows how to model equilibrium configurations of immiscible fluids using Fleming's flat chains with coefficients in a group. Existence is proved and regularity is discussed briefly. AMS-TeX, dvi, ps.
Remark: When I wrote this paper, I was not aware that Ambrosio and Braides had proved a more general result by different methods. See their papers "Functionals defined on partitions in sets of finite perimeter I, II", J. Math Pures Appl(9) 69 (1990), no. 3, 285--305 and 307--333.
This paper proves some abstract stratification theorems (generalizing work of Almgren) for closed sets or more generally for upper semicontinuous functions. (Closed sets correspond to 0-1 valued upper semicontinuous functions.) The theorems can be applied to minimal varieties, mean curvature flows, and energy minimizing p-harmonic maps to give stratifications by tangent cone or blow-up type.
AMS-TeX,
dvi,
ps
Figures (.gif): 1, 2,
3.
We show that for every k > 2\pi, there is a embedded non-closed curve in R^{3} with total curvature k that bounds a soap-film like minimal variety. Conversely, every curve that bounds such a variety (or any stationary varifold) must have total curvature > 2\pi. paper.
This paper proves the title assertion and discusses its relevance for boundary branch points. AMS-TeX, dvi, ps.
A condensed version appeared in the Notices of the Amer. Math. Soc. 44(11) (December, 1997), 1451-1456.
Entire paper: pdf (adobe acrobat), dvi, ps.
The condensed version is available from the online AMS Notices.
This paper proves that such surfaces have no boundary branch points. AMS-TeX, dvi, ps.
We prove that the deformation procedure of Federer and Fleming gives good approximations to arbritrary flat chains, not just those of finite mass and boundary mass. This implies that, for arbitrary coefficient groups, flat chains of finite mass and finite size are rectifiable, and also that, for finite coefficient groups, flat chains supported in sets of finite Hausdorff measure (or even finite integral geometric measure) have finite mass. dvi, ps.
We prove (without using the Besicovitch-Federer structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite-mass flat chain to be rectifiable.
The condition is the following: the group, regarded as a metric space, should contain no continuous, non-constant path of finite length. The p-adic numbers are an example of such a group. dvi, ps.
We prove that Federer's structure theorem for k-dimensional sets in euclidean N-space follows from the special case of 1-dimensional sets in the plane, which was proved earlier by Besicovitch. dvi, ps.
We prove that when a compact mean-convex subset of R^{n+1} (or of an (n+1)-dimensional riemannian manifold) moves by mean-curvature, the spacetime singular set has parabolic hausdorff dimension at most n-1. Examples show that this is optimal. We also show that, as t goes to infinity, the surface converges to a compact stable minimal hypersurface whose singular set has dimension at most n - 7. If n < 7, the convergence is everywhere smooth and hence after some time T, the moving surface has no singularities.
Consider the mean curvature flow generated by the boundary of compact mean-convex region of R^{n+1} or of an (n+1)-dimensional riemannian manifold. If n < 7, we prove that the moving boundary is very nearly convex in a spacetime neighborhood of any singularity. In particular, the tangent flows at singular points are all shrinking spheres or shrinking cylinders. If n>7, we prove the same result up to the first time that singularities occur. (In another paper, we will show that this is true even after that time, provided the initial surface is smooth.) dvi, pdf, ps.
This paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the Gaussian densities are close to 1. dvi, pdf, ps.
(with Tobias Ekholm and Daniel Wienholtz)
This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4 pi must be smoothly embedded. Related results are proved for varifolds and for soap film surfaces. dvi, ps.