Research Papers

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. Skew loops and quadric surfaces, with B. Solomon, Comment. Math. Helv., 77 (2002) 767-782.

    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.

  7. Circles minimize most knot energies, with A. Abrams, J. Cantarella, J. Fu, and R. Howard, Topology, 42 (2003) 381-394.

    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.

  8. The convex hull property and topology of hypersurfaces with nonnegative curvature, with S. Alexander, Adv. Math., 180 (2003), 324-354.

    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.

  9. 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.

  10. 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.

  11. 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.

  12. The convex hull property of noncompact hypersurfaces with positive curvature, with S. Alexander, Amer. J. Math., 126 (2004), 891-897.

    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.

  13. 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.

  14. 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.

  15. Total positive curvature of hypersurfaces with convex boundary, with J. Choe and M. Ritore, J. Differential Geom., 72 (2006), 129-147.

    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.

  16. h-Principles for hypersurfaces with prescribed principal curvatures and directions , with M. Kossowski, Tran. Amer. Math. Soc., 358 (2006), 4379-4393.

    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.

  17. Relative isoperimetric inequality outside convex domains in Rn, with J. Choe and M. Ritore, Calc. Var. Partial Differential Equations, 29 (2007), 421-429.

    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.

  18. h-Principles for curves and knots of constant curvature, 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.

  19. 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.

  20. Totally skew embeddings of manifolds, with S. Tabachnikov, Math. Z., 258 (2008), 499-512.

    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.

  21. Topology of negatively curved real affine algebraic surfaces, with C. Connell, to appear in J. Reine Angew. Math.

    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.

  22. Topology of Riemannian submanifolds with prescribed boundary, with S. Alexander, and J. Wong, Submitted.

    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.

  23. Relative isometric embeddings of Riemannian manifolds, with R. Greene, Submitted.

    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.

Expository Papers

  1. Classical open problems in Differential Geometry

    A survey of some open problems involving curves and surfaces in Euclidean space.