Strictly convex submanifolds and hypersurfaces of positive curvature,
J. Differential Geom., 57 (2001) 239-271.
We construct smooth closed hypersurfaces of positive curvature with prescribed submanifolds and tangent planes. Further, we develop some applications to boundary value problems via Monge-Ampere equations, smoothing of convex polytopes, and an extension of Hadamard's ovaloid theorem to hypersurfaces with boundary. In particular, we obtain an optimal regularity result for the boundary of convex hulls of submanifolds.
Gauss map, topology, and convexity of hypersurfaces with nonvanishing curvature,
Topology, 41 (2002) 107-117.
We prove that every immersion of a compact connected n-manifold into a sphere of the same dimension is an embedding, if it is one-to-one on each boundary component of the manifold. Some applications of this result are discussed for studying geometry and topology of hypersurfaces with non-vanishing curvature in Euclidean space, via their Gauss map; particularly, in relation to a conjecture of Meeks on minimal surfaces with convex boundary. It is also proved, as another application, that a compact hypersurface with nonvanishing curvature is convex, if its boundary lies in a hyperplane.
Solution to the shadow problem in 3-space,
in Minimal Surfaces, Geometric Analysis and Symplectic Geometry,
Adv. Stud. Pure Math, 34 (2002) 129-142.
If a convex surface, such as an egg shell, is illuminated from any given direction, then the corresponding shadow cast on the surface forms a connected subset. The shadow problem, first studied by H. Wente in 1978, asks whether a converse of this phenomenon is true as well. In this report it is shown that the answer is yes provided that each shadow is simply connected; otherwise, the answer is no. Further, the motivations behind this problem, and some ramifications of its solution for studying constant mean curvature surfaces in 3-space (soap bubbles) are discussed.
The problem of optimal smoothing for convex functions,
Proc. Amer. Math. Soc., 130 (2002) 2255-2259.
A procedure is described for smoothing a convex function which not only preserves its convexity, but also, under suitable conditions, leaves the function unchanged over nearly all the regions where it is already smooth. The method is based on a convolution followed by a gluing. Controlling the Hessian of the resulting function is the key to this process, and it is shown that it can be done successfully provided that the original function is strictly convex over the boundary of the smooth regions.
Shadows and convexity of surfaces,
Ann. of Math., 155 (2002) 281-293.
We study the geometry and topology of immersed surfaces in Euclidean 3-space whose Gauss map satisfies a certain two-piece-property, and solve the ``shadow problem" formulated by H. Wente. Also, we produce a counterexample to a conjecture of J. Choe.
Skew loops and quadric surfaces,
Comment. Math. Helv., 77 (2002) 767-782 (with B. Solomon).
A skew loop is an immersed circle in Euclidean space with no pair of parallel tangent lines. We prove that quadric surfaces of positive curvature--ellipsoids, elliptic paraboloids, and hyperboloids of two sheets--admit no such curves. Further, we show that this property characterizes the positively curved quadrics amongst all complete surfaces with at least one point of positive curvature immersed in 3-space. In particular, ellipsoids are the only closed surfaces without skew loops.
Circles minimize most knot energies,
Topology, 42 (2003) 381-394, (with A. Abrams, J. Cantarella, J. Fu, and R. Howard).
We define a new class of knot energies (called renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The proof is based on a theorem of Luko on average chord lengths of closed curves.
The convex hull property and topology of hypersurfaces
with nonnegative curvature,
Adv. Math., 180 (2003), 324-354, (with S. Alexander).
We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed hypersurface lies outside the convex hull of its boundary, provided that the boundary satisfies certain required conditions. This is a new convex hull property, dual to the classical one for surfaces with nonpositive curvature. Furthermore, we show that our boundary conditions determine the topology of the hypersurface up to at most two choices. Analogous results are obtained in the nonsmooth category. The proof uses uniform estimates for radii of convexity of locally convex hypersurfaces under a clipping procedure, together with a general convergence theorem.
A smooth convex loop with vanishing projections,
Topology, 43 (2004), 245.
We show that there exists a smooth convex simple closed curve in 3-space whose planar projections in 3 linearly independent directions do not bound any areas. This settles a problem ("the simple loop conjecture") which had been studied by Bruce Solomon.
Optimal smoothing for convex polytopes,
Bull. London. Math. Soc., 36 (2004), 483-492.
We prove that given a convex polytope P, together with a collection of compact convex subsets in the interior of each facet of P, there exists a smooth convex body arbitrarily close to P which coincides with each facet precisely along the prescribed sets, and has positive curvature elsewhere.
Shortest periodic billiard trajectories
in convex bodies,
Geom. Funct. Anal., 14 (2004), 295-302.
Motivated by applications to inverse spectral problems (``can one hear the shape of a drum?"), S. Zelditch has recently raised the question of whether every shortest periodic billiard trajectory in a bi-axisymmetric convex planar body is a bouncing ball (2-link) orbit. We show that the answer is yes by proving that the length of periodic billiard trajectories in any convex planar body K is at least 4 times the inradius of K; the equality holds precisely when the width of K is twice its inradius, in which case we show that the shortest periodic trajectories are all bouncing ball orbits.
The convex hull property of noncompact hypersurfaces
with positive curvature,
Amer. J. Math., 126 (2004), 891-897, (with S. Alexander).
We prove that in Euclidean n-space, every metrically complete, positively curved immersed hypersurface M, with compact boundary, lies outside the convex hull of its boundary provided that its boundary is embedded on a convex body, and n>2. For n=2, on the other hand, we construct examples which contradict this property.
Nonexistence of skew loops on ellipsoids,
Proc. Amer. Math. Soc., 133 (2005), 3687-3690.
We prove that every C1 closed curve immersed on an ellipsoid has a pair of parallel tangent lines. This establishes the optimal regularity for a phenomenon first observed by B. Segre. Our proof uses an approximation argument with the aid of an estimate for the size of loops in the tangential spherical image of a spherical curve.
Tangent bundle embeddings of manifolds in Euclidean space ,
Comment. Math. Helv., 81(2006), 259-270.
For a given n-manifold M we study the problem of finding the smallest integer N(M) such that M admits a smooth embedding in the N-dimensional Euclidean space without intersecting tangent spaces. We use the Poincare-Hopf index theorem to prove that N(S1)=4, and construct explicit examples to show that N(Sn)< 3n+4, where Sn denotes the n-sphere. Finally, for any closed n-manifold M, we show that 2n< N(Mn)< 4n+2.
Total positive curvature of hypersurfaces with convex boundary,
J. Differential Geom., 72 (2006), 129-147, (with J. Choe and M. Ritore).
We prove that if the boundary of a compact hypersurface in Euclidean n-space lies on the boundary of a convex body and meets that convex body orthogonally from the outside, then the total positive curvature of the hypersurface is bigger than or equal to half the area of the (n-1)-sphere . Also we obtain necessary and sufficient conditions for the equality to hold.
h-Principles for hypersurfaces with prescribed principal curvatures and directions ,
Tran. Amer. Math. Soc., 358 (2006), 4379-4393, (with M. Kossowski).
We prove that any compact orientable hypersurface with boundary immersed (resp. embedded) in Euclidean space is regularly homotopic (resp. isotopic) to a hypersurface with principal directions which may have any prescribed homotopy type, and principal curvatures each of which may be prescribed to within an arbitrary small error of any constant. Further we construct regular homotopies (resp. isotopies) which control the principal curvatures and directions of hypersurfaces in a variety of ways. These results, which we prove by holonomic approximation, establish some h-principles in the sense of Gromov, and generalize theorems of Gluck and Pan on embedding and knotting of positively curved surfaces in 3-space.
Relative isoperimetric inequality outside convex domains in Rn,
Calc. Var. Partial Differential Equations, 29 (2007), 421-429, (with J. Choe and M. Ritore).
We prove that the area of a hypersurface which traps a given volume outside of a convex body in Euclidean n-space must be greater than or equal to the area of a hemisphere trapping the given volume on one side of a hyperplane.
h-Principles for curves and knots of
Geom. Dedicata, 127 (2007), 19-35.
We prove that smooth (C∞) curves of constant curvature satisfy, in the sense of Gromov, the relative C1-dense h-principle in the space of immersed curves. In particular, in the isotopy class of any given C1 knot f in Euclidean space Rn≥ 3 there exists a smooth knot g of constant curvature which is C1-close to f. Further we show that if f is C2, then the curvature of g may be set equal to any constant c which is equal to or bigger than the maximum curvature of f. Furthermore, we may require that g be tangent to f along any finite set of prescribed points, and coincide with f over any compact set with an open neighborhood where f has constant curvature c. The proof involves some basic convexity theory and a sharp estimate for the position of the average value of a parameterized curve within its convex hull.
Topology of surfaces with connected shades,
Asian J. Math., 11 (2007), 621-634.
We prove that any closed orientable two dimensional manifold may be smoothly embedded in Euclidean 3-space so as to have connected shades (a.k.a. shadows) with respect to all directions of illumination.
Totally skew embeddings of manifolds,
Math. Z., 258 (2008), 499-512, (with S. Tabachnikov).
We study a version of Whitney's embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an n-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generalized vector field problem, existence of non-singular bilinear maps, and the immersion problem for real projective spaces. We use these connections and other methods to obtain several specific and general bounds for the desired dimension.
Topology of negatively curved real affine algebraic surfaces,
J. Reine Angew. Math., 624 (2008), 1-26, (with C. Connell).
We find a quartic example of a smooth embedded negatively curved surface in R3 homeomorphic to a doubly punctured torus. This constitutes an explicit solution to Hadamard's problem on constructing complete surfaces with negative curvature and Euler characteristic in R3. Further we show that our solution has the optimal degree of algebraic complexity via a topological classification for smooth cubic surfaces with a negatively curved component in R3: any such component must either be topologically a plane or an annulus. In particular we prove that there exists no cubic solutions to Hadamard's problem.
Topology of Riemannian submanifolds with prescribed boundary,
Duke Math. J., 152 (2010), 533-565, (with S. Alexander, and J. Wong).
We prove that a smooth compact immersed submanifold of codimension 2 in Rn, n>2, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimenion is too high, or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem, and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman.
Relative isometric embeddings of Riemannian manifolds,
Tran. Amer. Math. Soc., 363 (2011), 63-73, (with R. Greene.)
We prove the existence of C1 isometric embeddings, and C∞ approximate isometric embeddings, of Riemannian manifolds into Euclidean space with prescribed values in a neighborhood of a point.
A Riemannian four vertex theorem for surfaces with boundary,
Proc. Amer. Math. Soc., 139 (2011), 293-303.
We prove that every metric of constant curvature on a compact surface M with boundary bdM induces at least four vertices, i.e., local extrema of geodesic curvature, on a connected component of bdM, if, and only if, M is simply connected. Indeed, when M is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant curvature on M which induce only two critical points of geodesic curvature on each component of bdM. With few exceptions, these metrics are obtained by removing the singularities and a perturbation of flat structures on closed surfaces.
Directed immersions of closed manifolds,
Geom. Topol., 15 (2011) 699-705.
Given any finite subset X of the sphere Sn, n > 1, which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in Euclidean space Rn+1 whose Gauss map misses X. In particular, this answers a question of M. Gromov.
Normal curvatures of asymptotically constant graphs and Carathéodory's conjecture,
Proc. Amer. Math. Soc., 140 (2012), 4323-4335 (with R. Howard).
We show that Carathéodory's conjecture on umbilical points of closed convex surfaces may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R2 → R whose gradient decays uniformly faster than 1/r. The divergence theorem then yields a pair of integral equations for the normal curvatures of these graphs, which establish some weaker forms of the conjecture. In particular, we show that there are uncountably many principal lines in the graph of f whose projections into R2 are parallel to any given direction.
Deformations of unbounded convex bodies and hypersurfaces,
Amer. J. Math.,134 (2012),1585-1611.
We study the topology of the space ∂Kn of complete convex hypersurfaces of Rn which are homeomorphic to Rn-1. In particular, using Minkowski sums, we construct a deformation retraction of ∂Kn onto the Grassmannian space of hyperplanes. So every hypersurface in ∂Kn may be flattened in a canonical way. Further, the total curvature of each hypersurface evolves continuously and monotonically under this deformation. We also show that, modulo proper rotations, the subspaces of ∂Kn consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg.
Vertices of closed curves in Riemannian surfaces,
Comment. Math. Helv., 88 (2013), 427-448.
We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. Further we characterize the simply connected space forms as the only surfaces in which every closed curve bounding a compact immersed surface has more than two vertices.
Tangent lines, inflections, and vertices of closed curves,
Duke Math. J., 162 (2013), 2691-2730.
We show that every smooth closed curve C immersed in Euclidean space R3 satisfies the sharp inequality 2(P+I)+V ≥ 6 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. We also show that 2(P++I)+V≥ 4, where P+ is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane RP2 and the sphere S2 which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's ``tennis ball theorem".
Tangent cones and regularity of real hypersurfaces,
J. Reine Angew. Math., 697 (2014), 221-247 (with R. Howard).
We characterize C1 embedded hypersurfaces of Rn as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most m < 3/2. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is C1. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface X of Rn is C1. Furthermore, if X is real algebraic, strictly convex, and unbounded then its projective closure is a C1 hypersurface as well, which shows that X is the graph of a function defined over an entire hyperplane.
Affine unfoldings of convex polyhedra,
Geom. Topol., 18 (2014), 3055-3090.
We show that every convex polyhedron admits a simple edge unfolding after an affine transformation. In particular there exists no combinatorial obstruction to a positive resolution of Durer's unfoldability problem, which answers a question of Croft, Falconer, and Guy. Among other techniques, the proof employs a topological characterization for embeddings among the planar immersions of the disk.
Total diameter and area of closed submanifolds,
Math. Ann., 363 (2015), 985-999 (with R. Howard).
The total diameter of a closed planar curve C is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of C. Furthermore, when C is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when C is a circle. We also generalize these results to m dimensional submanifolds of Rn, where the "area" will be defined in terms of the mod 2 winding numbers of the submanifold about the n-m-1 dimensional affine subspaces of Rn.
Boundary torsion and convex caps of locally convex surfaces,
J. Differential Geom., 105 (2017), 427-486.
We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least 4 times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the 4 vertex theorem of Sedykh for convex space curves, and thus constitutes a far reaching extension of the classical 4 vertex theorem. The proof involves studying the arrangement of convex caps in a locally convex surface, and yields a Bose type formula for these objects.
Dürer's unfolding problem for convex polyhedra, Notices Amer. Math. Soc., 65 (2018), 25-27.
This is an expository article on some recent developments concerning the problem of Dürer on unfoldability of convex polyhedra, together with some historic background.
The length, width, and inradius of space curves, to appear in Geom. Dedicata.
The width w of a curve c in Euclidean space Rn is the infimum of the distances between all pairs of parallel hyperplanes which bound c, while its inradius r is the supremum of the radii of all spheres which are contained in the convex hull of c and are disjoint from c. We use a mixture of topological and integral geometric techniques, including an application of Borsuk Ulam theorem due to Wienholtz and Crofton's formulas, to obtain lower bounds on the length of c subject to constraints on r and w. The special case of closed curves is also considered in each category. Our estimates confirm some conjectures of Zalgaller up to 99% of their stated value, while we also disprove one of them.
Torsion of locally convex curves, submitted.
We show that the torsion of any simple closed curve C in Euclidean 3-space changes sign at least 4 times provided that it is star-shaped and locally convex with respect to a point o in the interior of its convex hull. The latter condition means that through each point p of C there passes a plane H, not containing o, such that a neighborhood of p in C lies on the same side of H as does o. This generalizes the four vertex theorem of Sedykh for convex space curves. Following Thorbergsson and Umehara, we reduce the proof to the result of Segre on inflections of spherical curves, which is also known as Arnold's tennis ball theorem.
Centers of disks in Riemannian manifolds (with I. Belegradek), submitted.
We prove the existence of a center, or continuous selection of a point, in the relative interior of C1 embedded k-disks in Riemannian n-manifolds. If k ≤ 3 the center can be made equivariant with respect to the isometries of the manifold, and under mild assumptions the same holds for k = 4 = n . By contrast, for every n ≥ k ≥ 6 there are examples where an equivariant center does not exist. The center can be chosen to agree with any of the classical centers defined on the set of convex compacta in the Euclidean space.
Nonnegatively curved hypersurfaces with free boundary on a sphere (with C. Xiong),
We prove that in Euclidean space Rn+1 any compact immersed nonnegatively curved hypersurface M with free boundary on the sphere Sn is an embedded convex topological disk. In particular, when the mth mean curvature of M is constant, for any 1 ≤ m ≤ n, M is a spherical cap or an equatorial disk.
Pseudo-edge unfoldings of convex polyhedra (with N. Barvinok), submitted.
A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable. Thus Durer's conjecture does not hold for pseudo-edge unfoldings. The proof is based on a result of Pogorelov on convex caps with prescribed curvature, and an unfoldability criterion for almost flat convex caps due to Tarasov.
Rigidity of nonnegatively curved surfaces relative to a curve, (with J. Spruck) submitted.
We prove that any properly oriented C2,1 isometric immersion of a positively curved Riemannian surface M into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in M. A generalization of this result to nonnegatively curved surfaces is presented as well under suitable conditions on their parabolic points. Thus we obtain a local version of Cohn-Vossen's rigidity theorem for convex surfaces subject to a Dirichlet condition. The proof employs in part Hormander's unique continuation principle for elliptic PDEs. Our approach also yields a short proof of Cohn-Vossen's theorem via Hopf's maximum principle.
Open problems in geometry of curves and surfaces, in progress.
We collect dozens of well-known and not so well-known fundamental unsolved problems involving low dimensional submanifolds of Euclidean space. The list includes selections from differential geometry, Riemannian geometry, metric geometry, discrete or polyhedral geometry, geometric knot theory, theory of convex bodies, and integral geometry. The common thread through these selections are the simplicity and intuitive nature of the questions. Extensive bibliography and historical background are included for each set of problems.