KNOPF, DANIEL F

Dan Knopf

Associate Dean for Graduate Education, Professor
College of Natural Sciences, Department of Mathematics


Geometric Analysis, Differential Geometry, Geometric PDE

danknopf@austin.utexas.edu

Phone: 512-471-8131

Office Location
RLM 9.152

Postal Address
The University of Texas at Austin
College of Natural Sciences
1 University Station G2500
Austin, TX 78712

Ph.D., University of Wisconsin-Milwaukee (1999)

Research Interests

Geometric analysis, Differential geometry, Geometric partial differential equations.

I am a member of the Geometry research group in the UT-Austin Department of Mathematics. I also interact with our research groups in Partial Differential Equations and Topology.

 

As a geometric analyst, I primarily study geometric heat flows. These are partial differential equations and systems that are nonlinear relatives of the heat equation. Intuitively, one expects such flows to improve a given geometric object, evolving it towards an optimal or canonical structure. But because geometric flows have a diffusion-reaction structure, their solutions often develop singularities. For these flows to have successful geometric applications requires a deep understanding of how singularities form and of how solutions can be continued past them. So my research includes extensive asymptotic analysis of singularity formation as well as detailed studies of dynamical stability of special solutions.

Research, publications and preprints

Sphere bundles with 1/4-pinched fiberwise metrics. Coauthors: Thomas Farrell, Zhou Gang, and Pedro Ontaneda. (arXiv:1505.03773)

Ricci flow neckpinches without rotational symmetry. Coauthors: James Isenberg and Natasa Sesum. (arXiv:1312.2933)

Universality in mean curvature flow neckpinches. Coauthor: Zhou Gang. Duke Math. J. To appear. (arXiv:1308.5600)

Neckpinch dynamics of asymmetric surfaces evolving by mean curvature flow. Coauthors: Zhou Gang and Israel Michael Sigal. (arXiv:1109.0939)

Degenerate neckpinches in Ricci flow. Coauthors: Sigurd Angenent and James Isenberg. J. Reine Angew. Math. (Crelle) To appear. (arXiv:1208.4312)

Neckpinch dynamics for asymmetric surfaces evolving by mean curvature flow. Coauthors: Zhou Gang and Israel Michael Sigal.

Minimally invasive surgery for Ricci flow singularities. Coauthors: Sigurd Angenent and M. Cristina Caputo. J. Reine Angew. Math. (Crelle) 672 (2012) 39-87.

Formal matched asymptotics for degenerate Ricci flow neckpinches. Coauthors: Sigurd Angenent and James Isenberg.Nonlinearity 24 (2011), 2265-2280.

Cross curvature flow on a negatively curved solid torus. Coauthors: Jason Deblois and Andrea Young. Algebr. Geom. Topol. 10 (2010), 343-372.

Convergence and stability of locally RN-invariant solutions of Ricci flow. J. Geom. Anal. 19 (2009), no. 4, 817-846.

Estimating the trace-free Ricci tensor in Ricci flow. Proc. Amer. Math. Soc. 137 (2009), no. 9, 3099-3103.

Asymptotic stability of the cross curvature flow at a hyperbolic metric. Coauthor: Andrea Young. Proc. Amer. Math. Soc. 137 (2009), no. 2, 699-709.

Local monotonicity and mean value formulas for evolving Riemannian manifolds. Coauthors: Klaus Ecker, Lei Ni, and Peter Topping. J. Reine Angew. Math. (Crelle) 616 (2008), 89-130.

Precise asymptotics of the Ricci flow neckpinch. Coauthor: Sigurd Angenent. Comm. Anal. Geom. 15 (2007), no. 4, 773-844.

Linear stability of homogeneous Ricci solitons. Coauthors: Christine Guenther and James Isenberg. Int. Math. Res. Not. (2006), Article ID 96253, 30 pp.

Positivity of Ricci curvature under the Kaehler-Ricci flow. Commun. Contemp. Math. 8 (2006), no. 1, 123-133.

An example of neckpinching for Ricci flow on Sn+1. Coauthor: Sigurd Angenent. Math. Res. Lett. 11 (2004), no. 4, 493-518.

Rotationally symmetric shrinking and expanding gradient Kaehler-Ricci solitons. Coauthors: Mikhail Feldman and Tom Ilmanen. J. Differential Geom. 65 (2003), no. 2, 169-209.

A lower bound for the diameter of solutions to the Ricci flow with nonzero H1(Mⁿ;R). Coauthor: Tom Ilmanen. Math. Res. Lett. 10 (2003), no. 2, 161-168.

Hamilton's injectivity radius estimate for sequences with almost nonnegative curvature operators. Coauthors: Bennett Chow and Peng Lu. Comm. Anal. Geom. 10 (2002), no. 5, 1151-1180.

Stability of the Ricci flow at Ricci-flat metrics. Coauthors: Christine Guenther and James Isenberg. Comm. Anal. Geom. 10 (2002), no. 4, 741-777.

New Li-Yau-Hamilton inequalities for the Ricci flow via the space-time approach. Coauthor: Bennett Chow. J. Differential Geom. 60 (2002), no. 1, 1-51.

Quasi-convergence of model geometries under the Ricci flow. Coauthor: Kevin McLeod. Comm. Anal. Geom. 9 (2001), no. 4, 879-919.

Quasi-convergence of the Ricci flow. Comm. Anal. Geom. 8 (2000), no. 2, 375-391.

Books, surveys, and expository articles

Neckpinching for asymmetric surfaces moving by mean curvature. Nonlinear Evolution Problems. Mathematisches Forschungsinstitut Oberwolfach Report No. 26/2012. (DOI: 10.4171/OWR/2012/26)

The Ricci Flow: Techniques and Applications, Part IV: Long Time Solutions and Related Topics. Coauthors: Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Peng Lu, Feng Luo, and Lei Ni. Mathematical Surveys and Monographs. To appear.

The Ricci Flow: Techniques and Applications, Part III: Geometric-Analytic Aspects. Coauthors: Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Peng Lu, Feng Luo, and Lei Ni. Mathematical Surveys and Monographs, Vol. 163. American Mathematical Society, Providence, RI, 2010.

The Ricci Flow: Techniques and Applications, Part II: Analytic Aspects. Coauthors: Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Peng Lu, Feng Luo, and Lei Ni. Mathematical Surveys and Monographs, Vol. 144. American Mathematical Society, Providence, RI, 2008.

The Ricci Flow: Techniques and Applications, Part I: Geometric Aspects. Coauthors: Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Peng Lu, Feng Luo, and Lei Ni. Mathematical Surveys and Monographs, Vol. 135. American Mathematical Society, Providence, RI, 2007.

An introduction to the Ricci flow neckpinch. Geometric Evolution Equations. Edited by Shu-Cheng Chang, Bennett Chow, Sun-Chin Chu, and Chang-Shou Lin. Contemporary Mathematics. Vol. 367, 141-148. American Mathematical Society, Providence, RI. 2005.

The Ricci flow: An Introduction. Coauthor: Bennett Chow. Mathematical Surveys and Monographs, Vol. 110. American Mathematical Society, Providence, RI, 2004.

Singularity models for the Ricci flow: an introductory survey. Variational Problems in Riemannian Geometry: Bubbles, Scans and Geometric Flows. Edited by Paul Baird, Ahmad El Soufi, Ali Fardoun, and Rachid Regbaoui. Progress in Nonlinear Differential Equations and Their Applications, Vol. 59, 67-80. Birkhaeuser, Basel, 2004.

An injectivity radius estimate for sequences of solutions to the Ricci flow having almost nonnegative curvature operators. Coauthors: Bennett Chow and Peng Lu. Proceedings of ICCM 2001. Edited by Chang-Shou Lin, Lo Yang, and Shing-Tung Yau. New Studies in Advanced Mathematics, Vol. 4, 249-256. International Press, Somerville, MA, 2004.

  • Graduate School Diversity Mentoring Fellowship, University of Texas, 2011-12.
  • Frank E. Gerth III Faculty Fellowship, University of Texas, 2009-14.
  • Summer Research Assignment, University of Texas, 2005.
  • University of Wisconsin-Milwaukee Dissertation Fellowship, 1996-98.
  • Office of Naval Research Graduate Fellowship, 1993-96.