Research Papers (Mostly Pre-2000)

Embeddedness of minimal surfaces with total boundary curvature at most 4 pi.

(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.

A local regularity theorem for classical mean curvature flows.

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.

The nature of singularities in mean curvature flow of mean-convex surfaces.

Consider the mean curvature flow generated by the boundary of compact mean-convex region of Rⁿ⁺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.

The size of the singular set in mean curvature flow of mean-convex surfaces.

We prove that when a compact mean-convex subset of Rⁿ⁺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.

A new proof of Federer's structure theorem for k-dimensional sets in R^N.

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.

Rectifiability of flat chains

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 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.

This paper proves that such surfaces have no boundary branch points. AMS-TeX, dvi, ps.

Entire paper: pdf (adobe acrobat), dvi, ps.

The condensed version is available from the online AMS Notices.

This paper proves the title assertion and discusses its relevance for boundary branch points. AMS-TeX, dvi, ps.

Soap-films bounded by non-closed curves

We show that for every k > 2\pi, there is a embedded non-closed curve in R³ 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.

Stratification of minimal surfaces, mean curvature flows, and harmonic maps

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.

Existence of least-energy configurations of immiscible fluids

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.

The topology of hypersurfaces moving by mean curvature

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.

Some questions of De Giorgi about mean curvature flow of triply periodic surfaces

Certain triply periodic surfaces are shown to flow smoothly to triply periodic minimal surfaces, whereas others disappear in finite time.

Partial regularity of mean-convex hypersurfaces flowing by mean curvature

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.

The bridge principle for unstable and for singular minimal surfaces

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(B³) 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.

The bridge principle for stable minimal surfaces

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.

A strong minimax property of nondegenerate minimal submanifolds

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.

The structure of branch points in area minimizing surfaces and in pseudoholomorphic curves

If a minimal surface in R²ⁿ⁺²=Cⁿ⁺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 --> Cⁿ 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 C¹ 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 C¹ cannot be improved to C².

Nonunique tangent maps at isolated singularities of energy minimizing maps

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.

The Space of Minimal Annuli Bounded by an Extremal Pair of Planar Curves

Let G be a pair of convex plane curves in R³ 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.

Nonunique tangent maps at isolated singularities of harmonic maps

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.

Regularity of singular sets for plateau-type problems

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).

On the topological type of minimal submanifolds

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.

The space of minimal submanifolds for varying riemannian metrics

Let S be the space of ordered pairs where g is a riemannian metric on a (fixed) manifold N and M is a g-minimal compact submanifold of N. This paper shows that S is a Banach manifold, and uses the Banach manifold structure to prove certain existence results. For instance, it is shown that every metric on S³ near the standard metric admits at least four minimal embedded 2-spheres and at least five minimal embedded tori, and that every metric of positive Ricci curvature on S³ admits at least two minimal embedded 2-spheres and a minimal embedded torus.

Existence of smooth embedded surfaces of prescribed topological type that minimize parametric even elliptic functionals on three-manifolds

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.

Minimal Surfaces Bounded by Convex Curves in Parallel Planes

Let G be a pair of convex curves in parallel planes in R³. 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.

A rigidity theorem for properly embedded minimal surfaces in R³

Let f be a proper isometric minimal embedding in R³ of a complete surface M with more than one end. This paper shows that if g is any isometric minimal immersion of M in R³, then g = R f for some ambient isometry R: R³ --> R³.

Some Recent Developments in Differential Geometry

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.

The rate of convergence of a harmonic map at a singular point

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.)

Every three-sphere of positive ricci curvature contains a minimal embedded torus

This paper proves the title assertion.

A new proof of the compactness theorem for integral currents

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.)

New applications of mapping degrees to minimal surface theory

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.

A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals

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.

Homotopy classes in sobolev spaces and the existence of energy minimizing maps

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).

Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals

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.

Complete surfaces of finite total curvature

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 Rⁿ, 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.

The space of m-dimensional surfaces that are stationary for a parametric elliptic integrand

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 Rⁿ 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.

A regularity theorem for minimizing hypersurfaces modulo 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.

Infima of energy functionals in homotopy classes of mappings

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].

Homotopy classes in sobolev spaces and energy minimizing maps

Research announcement. See "Infima of energy functionals..." (J. Diff. Geom., 1986) and "Homotopy classes..." (Acta Math., 1988).

Regularity of singular sets in immiscible fluid interfaces and in solutions to other plateau-type problems

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.

Generic regularity of unoriented two-dimensional area minimizing surfaces

This paper shows that for almost every smooth closed curve in Rⁿ (or in a riemannian manifold), the unoriented area minimizing surfaces (i. e., flat chains mod 2) that it bounds have no singularities.

Mappings that minimize area in their homotopy classes

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.

The least area bounded by multiples of a curve

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 R⁴ such that

Existence of least-area mappings of N-dimensional domains

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.

Tangent cones to 2-dimensional area-minimizing integral currents are unique

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.

Regularity of area-minimizing hypersurfaces at boundaries with multiplicity

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 structure of minimizing hypersurfaces mod 4

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.