Research publications and preprints

Books, surveys, and expository articles

- Office location: PMA (RLM) 9.152
- Email address: danknopf {at} math {dot} utexas {dot} edu
- Phone number: 512.471.8131
- Fax number: 512.471.9038
- Office hours: By appointment.
- Mailing address:

The University of Texas at Austin

Mathematics Department, RLM 8.100

2515 Speedway, Stop C1200

Austin, TX 78712-1202

Mathematics Department, RLM 8.100

2515 Speedway, Stop C1200

Austin, TX 78712-1202

- To contact me in my role as Associate Dean for Graduate Education in the College of Natural Sciences,

please email danknopf {at} austin {dot} utexas {dot} edu, or phone at 512.475.6418.

- Geometric analysis
- Differential geometry
- Geometric partial differential equations

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

Singularity formation of complete Ricci flow solutions. Coauthors: Timothy Carson, James Isenberg, and Natasa Sesum. Submitted. (arXiv:2001.06098)

Ricci solitons, conical singularities, and nonuniqueness. Coauthor: Sigurd Angenent. Submitted. (arXiv:1909.08087)

Non-Kaehler Ricci flow singularities modeled on Kaehler-Ricci solitons.
Coauthors: James Isenberg and Natasa Sesum. **Pure Appl. Math. Q. 15**
(2019), no. 2, 749-784.

Dynamic instability of **CP ^{N}** under Ricci flow.
Coauthor: Natasa Sesum.

Sphere Bundles with 1/2-pinched Fiberwise Metrics.
Coauthors: Thomas Farrell, Zhou Gang, and Pedro Ontaneda.
**Trans. Amer. Math. Soc. 369 ** (2017), no. 9, 6613-6630.

Ricci flow neckpinches without rotational symmetry.
Coauthors: James Isenberg and Natasa Sesum.
**Comm. Partial Differential Equations 41** (2016), no. 12, 1860-1894.

Universality in mean curvature flow neckpinches.
Coauthor: Zhou Gang.
** Duke Math. J. 164** (2015), no. 12, 2341-2406.

Neckpinch dynamics for asymmetric surfaces evolving by mean curvature flow.
Coauthors: Zhou Gang and Israel Michael Sigal.
**Mem. Amer. Math. Soc. 253 ** (2018), no. 1210, 1-78.

Degenerate neckpinches in Ricci flow.
Coauthors: Sigurd Angenent and James Isenberg.
**J. Reine Angew. Math. (Crelle) 709** (2015), 81-117.

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 **R ^{N}**-invariant solutions of Ricci flow.

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 096253.

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 **S ^{n+1}**
Coauthor: Sigurd Angenent.

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 H^{1}(M;**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.

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,* Vol. 206. American Mathematical Society, Providence, RI, 2015.

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.

M310P – Plan II Mathematics: Through the Lens of Mathematics (Fall 2019)

M310P – Plan II Mathematics: Through the Lens of Mathematics (Fall 2018)

M310P – Plan II Mathematics: Through the Lens of Mathematics (Fall 2017)

M310P – Plan II Mathematics: Through the Lens of Mathematics (Fall 2016)

M427J – Differential Equations with Linear Algebra – *Math Honors* (Spring 2016)

M310P – Plan II Mathematics: Through the Lens of Mathematics (Fall 2015)

M310P - Plan II Mathematics: Through the Lens of Mathematics (Fall 2014)

M427K – Advanced Calculus for Applications I – *Math Honors* (Spring 2014)

M427K – Advanced Calculus for Applications I (Fall 2013)

M427K – Advanced Calculus for Applications I – *Math Honors* (Spring 2013)

TC310 – Plan II Modes of Reasoning: Optimal Geometry in Nature, Art, and Mathematics (Fall 2012)

M427K – Advanced Calculus for Applications I – *Math Honors* (Spring 2012)

M427K – Advanced Calculus for Applications I (Fall 2011)

M408C – Differential and Integral Calculus (Spring 2011)

M392C – Riemannian Geometry (Fall 2010)

TC310 – Plan II Modes of Reasoning: Optimal Geometry in Nature, Art, and Mathematics (Spring 2010)

M408C – Differential and Integral Calculus (Fall 2009)

M427K – Advanced Calculus for Applications I -* Engineering Honors* (Spring 2009)

M365G – Curves and Surfaces (Spring 2009)

M427K – Advanced Calculus for Applications I (Fall 2008)

M382D – Differential Topology (Spring 2008)

Tim Carson (PhD, May 2018)

Haotian Wu (PhD, May 2013)

Davi Maximo (PhD, May 2013)

Michael Bradford Williams (PhD, May 2011)

Bradley Anderson (MA, May 2008)

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