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Locally Convex Hypersurfaces

• Strictly convex submanifolds and hypersurfaces of positive curvature,
  J. Differential Geom., 57 (2001) 239-271.
• The convex hull property and topology of hypersurfaces with nonnegative curvature,
  Adv. Math., 180 (2003), 324-354 (with S. Alexander).
• The convex hull property of noncompact hypersurfaces with positive curvature,
  Amer. J. Math., 126 (2004), 891-897 (S. Alexander).
• Total positive curvature of hypersurfaces with convex boundary,
  J. Differential Geom., 72 (2006), 129-147 (with J. Choe and M. Ritore).
• Topology of Riemannian submanifolds with prescribed boundary,
  Duke Math. J., 152 (2010), 533-565 (with S. Alexander and J. Wong).
• Deformations of unbounded convex bodies and hypersurfaces,
  Amer. J. Math.,134 (2012), 1585-1611.
• Tangent cones and regularity of real hypersurfaces,
  J. Reine Angew. Math., 697 (2014), 221-247 (with R. Howard).
• Boundary torsion and convex caps of locally convex surfaces,
  J. Differential Geom., 105 (2017), 427-486.
• Nonnegatively curved hypersurfaces with free boundary on a sphere,
  Calc. Var. Partial Differential Equations (with C. Xiong) 58 (2019), Art. 94, 20 pp.
• Rigidity of nonnegatively curved surfaces relative to a curve,
  Int. Math. Res. Not. IMRN. (with J. Spruck), 17 (2020), 5387-5400.
• Rigidity of nonpositively curved manifolds with convex boundary,
  Proc. Amer. Math. Soc., 151 (2023), 4935-4940 (with J. Spruck).
• Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds,
  submitted.

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The h-Principle and Isometric Embeddings

• h-Principles for hypersurfaces with prescribed principal curvatures and directions,
  Tran. Amer. Math. Soc., 358 (2006), 4379-4393 (with M. Kossowski).
• h-Principles for curves and knots of constant curvature,
  Geom. Dedicata, 127 (2007), 19-35.
• Directed immersions of closed manifolds,
  Geom. Topol., 15 (2011) 699-705.
• Relative isometric embeddings of Riemannian manifolds,
  Tran. Amer. Math. Soc., 363 (2011), 63-73 (with R. Greene).
• Rigidity of nonnegatively curved surfaces relative to a curve,
  Int. Math. Res. Not. IMRN. (with J. Spruck), 17 (2020), 5387-5400.
• Deformations of curves with constant curvature,
  submitted (with M. Raffaelli).
• h-Principles for curves and knots of constant torsion,
  submitted (with M. Raffaelli).
• Topology of closed asymptotic curves on negatively curved surfaces,
  preprint (with M. Raffaelli).

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Isoperimetric and Other Geometric Inequalities

• Circles minimize most knot energies,
  Topology, 42 (2003) 381-394 (with A. Abrams, J. Cantarella, J. Fu, and R. Howard).
• Total positive curvature of hypersurfaces with convex boundary,
  J. Differential Geom., 72 (2006), 129-147 (with J. Choe and M. Ritore).
• Relative isoperimetric inequality outside convex domains in Rⁿ,
  Calc. Var. Partial Differential Equations, 29 (2007), 421-429 (with J. Choe and M. Ritore).
• Total diameter and area of closed submanifolds,
  Math. Ann., 363 (2015), 985-999 (with R. Howard).
• The length, width, and inradius of space curves,
  Geom. Dedicata, 196 (2018), 123-143.
• Shortest closed curve to inspect a sphere,
  J. Reine Angew. Math. (with J. Wenk), 781 (2021), 57-84.
• Total curvature and the isoperimetric inequality in Cartan-Hadamard manifolds,
  J. Geom. Anal. 32 (2022), Paper No. 50, 54pp. (with J. Spruck).
• Total mean curvatures of Riemannian hypersurfaces,
  Adv. Nonlinear Stud., 23 (2023) pp. 20220029 (with J. Spruck).
• Rigidity of nonpositively curved manifolds with convex boundary,
  Proc. Amer. Math. Soc., 151 (2023), 4935-4940 (with J. Spruck).
• Minkowski inequality in Cartan-Hadamard manifolds,
  Int. Math. Res. Not. IMRN., 20 (2023), 17892-17910 (with J. Spruck).
• Shortest closed curve to contain a sphere in its convex hull,
  to appear in Bull. London. Math. Soc. (with J. Wenk).
• Comparison formulas for total mean curvatures of Riemannian hypersurfaces,
  to appear in Adv. Nonlinear Stud.
• Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds,
  submitted.

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Spherical Images and Tangent Lines of Submanifolds

• Gauss map, topology, and convexity of hypersurfaces with nonvanishing curvature,
  Topology, 41 (2002) 107-117.
• Solution to the shadow problem in 3-space, in Minimal Surfaces, Geometric Analysis and Symplectic Geometry, Adv. Stud. Pure Math, 34 (2002) 129-142.
• Shadows and convexity of surfaces,
  Ann. of Math., 155 (2002) 281-293.
• Skew loops and quadric surfaces,
  Comment. Math. Helv., 77 (2002) 767-782 (with B. Solomon).
• Nonexistence of skew loops on ellipsoids,
  Proc. Amer. Math. Soc., 133 (2005), 3687-3690.
• Tangent bundle embeddings of manifolds in Euclidean space ,
  Comment. Math. Helv., 81 (2006), 259-270.
• Topology of surfaces with connected shades,
  Asian J. Math., 11 (2007), 621-634.
• Totally skew embeddings of manifolds,
  Math. Z., 258 (2008), 499-512 (with S. Tabachnikov).
• Tangent lines, inflections, and vertices of closed curves,
  Duke Math. J., 162 (2013), 2691-2730.

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Discrete and Polyhedral Geometry

• Shortest periodic billiard trajectories in convex bodies,
  Geom. Funct. Anal., 14 (2004), 295-302.
• Optimal smoothing for convex polytopes,
  Bull. London. Math. Soc., 36 (2004), 483-492.
• Affine unfoldings of convex polyhedra,
  Geom. Topol., 18 (2014), 3055-3090.
• Dürer's unfolding problem for convex polyhedra,
  Notices Amer. Math. Soc., 65 (2018), 25-27.
• Pseudo-edge unfoldings of convex polyhedra,
  Discrete Comput. Geom. (with N. Barvinok), 64 (2020), 671-689.
• 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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Special Points of Closed Curves and Surfaces

• A Riemannian four vertex theorem for surfaces with boundary,
  Proc. Amer. Math. Soc., 139 (2011), 293-303.
• Vertices of closed curves in Riemannian surfaces,
  Comment. Math. Helv., 88 (2013), 427-448.
• Normal curvatures of asymptotically constant graphs and Carathéodory's conjecture,
  Proc. Amer. Math. Soc., 140 (2012), 4323-4335 (with R. Howard).
• Tangent lines, inflections, and vertices of closed curves,
  Duke Math. J., 162 (2013), 2691-2730.
• Boundary torsion and convex caps of locally convex surfaces,
  J. Differential Geom., 105 (2017), 427-486.
• Torsion of locally convex curves,
  Proc. Amer. Math. Soc., 147 (2019), 1699-1707.
• Topology of closed asymptotic curves on negatively curved surfaces,
  preprint (with M. Raffaelli).

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Real Algebraic Geometry

• Topology of negatively curved real affine algebraic surfaces,
  J. Reine Angew. Math., 624 (2008), 1-26 (with C. Connell).
• Tangent cones and regularity of real hypersurfaces,
  J. Reine Angew. Math., 697 (2014), 221-247 (with R. Howard).

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Other

• The problem of optimal smoothing for convex functions,
  Proc. Amer. Math. Soc., 130 (2002) 2255-2259.
• A smooth convex loop with vanishing projections,
  Topology, 43 (2004), 245.
• Centers of disks in Riemannian manifolds,
  Pacific J. Math. (with I. Belegradek), 304 (2020), 401-418.
• Point selections from Jordan domains in Riemannian surfaces,
  to appear in Tran. Amer. Math. Soc. (with I. Belegradek).

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Expository

• Dürer's unfolding problem for convex polyhedra,
  Notices Amer. Math. Soc., 65 (2018), 25-27.
• Solution to the shadow problem in 3-space, in Minimal Surfaces, Geometric Analysis and Symplectic Geometry, Adv. Stud. Pure Math, 34 (2002) 129-142.
• Open problems in geometry of curves and surfaces, in progress.