Locally Convex Hypersurfaces, Nonnegative Curvature

Strictly convex submanifolds and hypersurfaces of positive curvature,
J. Differential Geom., 57 (2001) 239271.

The convex hull property and topology of hypersurfaces
with nonnegative curvature,
Adv. Math., 180 (2003), 324354 (with S. Alexander).

The convex hull property of noncompact hypersurfaces
with positive curvature,
Amer. J. Math., 126 (2004), 891897 (S. Alexander).

Total positive curvature of hypersurfaces with convex boundary,
J. Differential Geom., 72 (2006), 129147 (with J. Choe and
M. Ritore).

Topology of Riemannian submanifolds with prescribed boundary,
Duke Math. J., 152 (2010), 533565 (with
S. Alexander and J. Wong).

Deformations of unbounded convex bodies and hypersurfaces,
Amer. J. Math.,134 (2012), 15851611.

Tangent cones and regularity of real hypersurfaces,
J. Reine Angew. Math., 697 (2014), 221247 (with R. Howard).

Boundary torsion and convex caps of locally convex surfaces,
J. Differential Geom.,
105 (2017), 427486.

Nonnegatively curved hypersurfaces with free boundary on a sphere
(with C. Xiong), submitted.

Rigidity of nonnegatively curved surfaces relative to a curve, (with J. Spruck) submitted.
The hPrinciple, Convex Integration, and Isometric Embeddings

hPrinciples for hypersurfaces with prescribed principal curvatures and directions,
Tran. Amer. Math. Soc., 358 (2006), 43794393
(with M. Kossowski).

hPrinciples for curves and knots of constant curvature,
Geom. Dedicata, 127 (2007), 1935.

Directed immersions of closed manifolds,
Geom. Topol., 15 (2011) 699705.

Relative isometric embeddings of Riemannian manifolds,
Tran. Amer. Math. Soc., 363 (2011), 6373 (with
R. Greene).

Rigidity of nonnegatively curved surfaces relative to a curve, (with J. Spruck) preprint.
Spherical Images of Surfaces

Gauss map, topology, and convexity of hypersurfaces with nonvanishing curvature,
Topology,
41 (2002) 107117.

Solution to the shadow problem in 3space,
in Minimal Surfaces, Geometric Analysis and Symplectic Geometry,
Adv. Stud. Pure Math, 34 (2002) 129142.

Shadows and convexity of surfaces,
Ann. of Math.,
155 (2002) 281293.

Topology of surfaces with connected shades,
Asian J. Math., 11 (2007), 621634.
Skew Submanifolds and Quadric Surfaces

Skew loops and quadric surfaces,
Comment. Math. Helv., 77 (2002) 767782 (with
B. Solomon).

Nonexistence of skew loops on ellipsoids,
Proc. Amer. Math. Soc.,
133 (2005), 36873690.

Tangent bundle embeddings of manifolds in Euclidean space ,
Comment. Math. Helv., 81 (2006), 259270.

Totally skew embeddings of manifolds,
Math. Z., 258 (2008), 499512 (with
S. Tabachnikov).

Tangent lines, inflections, and vertices of closed curves,
Duke Math. J., 162 (2013), 26912730.
Discrete and Polyhedral Geometry

Shortest periodic billiard trajectories
in convex bodies,
Geom. Funct. Anal., 14 (2004), 295302.

Optimal smoothing for convex polytopes,
Bull. London. Math. Soc., 36 (2004), 483492.

Affine unfoldings of convex polyhedra,
Geom. Topol., 18 (2014), 30553090.

Dürer's unfolding problem for convex polyhedra,
Notices Amer. Math. Soc., 65 (2018), 2527.

Pseudoedge unfoldings of convex polyhedra (with N. Barvinok), submitted.
Isoperimetric and Other Geometric Inequalities

Circles minimize most knot energies,
Topology, 42 (2003) 381394 (with A. Abrams,
J. Cantarella,
J. Fu, and
R. Howard).

Total positive curvature of hypersurfaces with convex boundary,
J. Differential Geom., 72 (2006), 129147 (with J. Choe and
M. Ritore).

Relative isoperimetric inequality outside convex domains in R^{n},
Calc. Var. Partial Differential Equations, 29 (2007), 421429 (with J. Choe and
M. Ritore).

Total diameter and area of closed submanifolds,
Math. Ann., 363 (2015), 985999 (with R. Howard).

The, length, width, and inradius of space curves,
to appear in Geom. Dedicata.
Special Points of Closed Curves and Surfaces

A Riemannian four vertex theorem for surfaces with boundary,
Proc. Amer. Math. Soc., 139 (2011), 293303.

Vertices of closed curves in Riemannian surfaces,
Comment. Math. Helv., 88 (2013), 427448.

Normal curvatures of asymptotically constant graphs and Carathéodory's conjecture,
Proc. Amer. Math. Soc., 140 (2012), 43234335 (with R. Howard).

Tangent lines, inflections, and vertices of closed curves,
Duke Math. J., 162 (2013), 26912730.

Boundary torsion and convex caps of locally convex surfaces,
J. Differential Geom.,
105 (2017), 427486.

Torsion of locally convex curves, submitted.
Real Algebraic Geometry
Other
Expository