Locally Convex Hypersurfaces
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Strictly convex submanifolds and hypersurfaces of positive curvature,
J. Differential Geom., 57 (2001) 239-271.
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The convex hull property and topology of hypersurfaces
with nonnegative curvature,
Adv. Math., 180 (2003), 324-354 (with S. Alexander).
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The convex hull property of noncompact hypersurfaces
with positive curvature,
Amer. J. Math., 126 (2004), 891-897 (S. Alexander).
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Total positive curvature of hypersurfaces with convex boundary,
J. Differential Geom., 72 (2006), 129-147 (with J. Choe and
M. Ritore).
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Topology of Riemannian submanifolds with prescribed boundary,
Duke Math. J., 152 (2010), 533-565 (with
S. Alexander and J. Wong).
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Deformations of unbounded convex bodies and hypersurfaces,
Amer. J. Math.,134 (2012), 1585-1611.
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Tangent cones and regularity of real hypersurfaces,
J. Reine Angew. Math., 697 (2014), 221-247 (with R. Howard).
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Boundary torsion and convex caps of locally convex surfaces,
J. Differential Geom.,
105 (2017), 427-486.
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Nonnegatively curved hypersurfaces with free boundary on a sphere,
Calc. Var. Partial Differential Equations (with C. Xiong) 58 (2019), Art. 94, 20 pp.
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Rigidity of nonnegatively curved surfaces relative to a curve,
Int. Math. Res. Not. IMRN. (with J. Spruck), 17 (2020), 5387-5400.
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Rigidity of nonpositively curved manifolds with convex boundary,
Proc. Amer. Math. Soc., 151 (2023), 4935-4940 (with J. Spruck).
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Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds,
submitted.
The h-Principle and Isometric Embeddings
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h-Principles for hypersurfaces with prescribed principal curvatures and directions,
Tran. Amer. Math. Soc., 358 (2006), 4379-4393
(with M. Kossowski).
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h-Principles for curves and knots of constant curvature,
Geom. Dedicata, 127 (2007), 19-35.
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Directed immersions of closed manifolds,
Geom. Topol., 15 (2011) 699-705.
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Relative isometric embeddings of Riemannian manifolds,
Tran. Amer. Math. Soc., 363 (2011), 63-73 (with
R. Greene).
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Rigidity of nonnegatively curved surfaces relative to a curve,
Int. Math. Res. Not. IMRN. (with J. Spruck), 17 (2020), 5387-5400.
Isoperimetric and Other Geometric Inequalities
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Circles minimize most knot energies,
Topology, 42 (2003) 381-394 (with A. Abrams,
J. Cantarella,
J. Fu, and
R. Howard).
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Total positive curvature of hypersurfaces with convex boundary,
J. Differential Geom., 72 (2006), 129-147 (with J. Choe and
M. Ritore).
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Relative isoperimetric inequality outside convex domains in Rn,
Calc. Var. Partial Differential Equations, 29 (2007), 421-429 (with J. Choe and
M. Ritore).
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Total diameter and area of closed submanifolds,
Math. Ann., 363 (2015), 985-999 (with R. Howard).
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The length, width, and inradius of space curves,
Geom. Dedicata, 196 (2018), 123-143.
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Shortest closed curve to inspect a sphere,
J. Reine Angew. Math. (with J. Wenk), 781 (2021), 57-84.
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Total curvature and the isoperimetric inequality in Cartan-Hadamard manifolds,
J. Geom. Anal. 32 (2022), Paper No. 50, 54pp. (with J. Spruck).
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Total mean curvatures of Riemannian hypersurfaces,
Adv. Nonlinear Stud., 23 (2023) pp. 20220029 (with J. Spruck).
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Rigidity of nonpositively curved manifolds with convex boundary,
Proc. Amer. Math. Soc., 151 (2023), 4935-4940 (with J. Spruck).
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Minkowski inequality in Cartan-Hadamard manifolds,
Int. Math. Res. Not. IMRN., 20 (2023), 17892-17910 (with J. Spruck).
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Shortest closed curve to contain a sphere in its convex hull,
to appear in Bull. London. Math. Soc. (with J. Wenk).
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Comparison formulas for total mean curvatures of Riemannian hypersurfaces,
to appear in Adv. Nonlinear Stud.
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Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds,
submitted.
Spherical Images and Tangent Lines of Submanifolds
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Gauss map, topology, and convexity of hypersurfaces with nonvanishing curvature,
Topology,
41 (2002) 107-117.
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Solution to the shadow problem in 3-space,
in Minimal Surfaces, Geometric Analysis and Symplectic Geometry,
Adv. Stud. Pure Math, 34 (2002) 129-142.
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Shadows and convexity of surfaces,
Ann. of Math.,
155 (2002) 281-293.
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Skew loops and quadric surfaces,
Comment. Math. Helv., 77 (2002) 767-782 (with
B. Solomon).
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Nonexistence of skew loops on ellipsoids,
Proc. Amer. Math. Soc.,
133 (2005), 3687-3690.
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Tangent bundle embeddings of manifolds in Euclidean space ,
Comment. Math. Helv., 81 (2006), 259-270.
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Topology of surfaces with connected shades,
Asian J. Math., 11 (2007), 621-634.
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Totally skew embeddings of manifolds,
Math. Z., 258 (2008), 499-512 (with
S. Tabachnikov).
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Tangent lines, inflections, and vertices of closed curves,
Duke Math. J., 162 (2013), 2691-2730.
Discrete and Polyhedral Geometry
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Shortest periodic billiard trajectories
in convex bodies,
Geom. Funct. Anal., 14 (2004), 295-302.
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Optimal smoothing for convex polytopes,
Bull. London. Math. Soc., 36 (2004), 483-492.
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Affine unfoldings of convex polyhedra,
Geom. Topol., 18 (2014), 3055-3090.
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Dürer's unfolding problem for convex polyhedra,
Notices Amer. Math. Soc., 65 (2018), 25-27.
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Pseudo-edge unfoldings of convex polyhedra,
Discrete Comput. Geom. (with N. Barvinok), 64 (2020), 671-689.
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Shortest closed curve to contain a sphere in its convex hull,
to appear in Bull. London. Math. Soc. (with J. Wenk).
Special Points of Closed Curves and Surfaces
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A Riemannian four vertex theorem for surfaces with boundary,
Proc. Amer. Math. Soc., 139 (2011), 293-303.
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Vertices of closed curves in Riemannian surfaces,
Comment. Math. Helv., 88 (2013), 427-448.
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Normal curvatures of asymptotically constant graphs and Carathéodory's conjecture,
Proc. Amer. Math. Soc., 140 (2012), 4323-4335 (with R. Howard).
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Tangent lines, inflections, and vertices of closed curves,
Duke Math. J., 162 (2013), 2691-2730.
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Boundary torsion and convex caps of locally convex surfaces,
J. Differential Geom.,
105 (2017), 427-486.
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Torsion of locally convex curves,
Proc. Amer. Math. Soc., 147 (2019), 1699-1707.
Real Algebraic Geometry
Other
Expository