My main research interest is geometric analysis with emphasis currently on the theory of minimal and constant mean curvature surfaces, and mean curvature flow.
Minimal surfaces are defined as surfaces which are critical points for the area functional. The mean curvature of a minimal surface, the average of the two principal curvatures, is identically zero. Surfaces which are critical points for the area functional under a volume constraint are instead called constant mean curvature (CMC) surfaces and in fact the average of the two principal curvatures is constant. Not only are there plenty of mathematical examples for both of these surfaces (for instance planes, helicoids and catenoids are minimal surfaces while spheres, cylinders and Delaunay surfaces are non zero CMC surfaces), but they can easily be realized and observed in the real world. In nature, the shape of a soap film approximates with great accuracy that of a minimal surface while soap bubbles provide the analogous approximation for CMC surfaces. In other words, a soap bubble is the leastarea surface that encloses the fixed volume inside. Mean curvature flow is an example of geometric flows. When a surface moves under mean curvature flow then the normal component of the velocity at a point is given by the mean curvature of the surface at that point. The use of geometric flows such as mean curvature flow has been very fruitful in the study of a several important problems in differential geometry, image processing and mathematical physics, leading to a profound impact on each of these fields. They also arise very naturally in various physical contexts such as thermomechanics, annealing metals, crystal growth, flame propagation and wearing processes.
KCL/UCL Geometry Seminar
Information for potential applicants to our PhD program
Papers published or accepted for publication
My papers are listed below with a link to the journal where the PDF DOCUMENT can be downloaded. If you would like to have a copy of my paper but do not have access to the relevant journal, please send me an email at . Alternatively, most of my papers/preprints are posted on arXiv. Note however that the published version might be slightly different, more up to date, from the one on arXiv.

Triply periodic constant mean curvature surfaces. Advances in Mathematics, Volume 335, 7 September 2018, Pages 809837. PDF document

Constant mean curvature surfaces. Surveys in Differential Geometry, International Press, (2016). PDF document

Chord arc properties for constant mean curvature disks. Geometry & Topology 22 (2018) 305–322. PDF document

Nonproperly embedded $H$planes in $\mathbb{H}^2\times\mathbb{R}$. Math. Ann. (2018), 370(3), 14911512. PDF document

Nonproperly embedded $H$planes in $\mathbb{H}^3$. J. Differential Geometry, Volume 105, Number 3 (2017), 405425. PDF document

Topological Type of Limit Laminations of Embedded Minimal Disks. J. Differential Geometry, Volume 102, Number 1 (2016), 123. PDF document

Nonproper complete minimal surfaces embedded in $\mathbb{H}^2\times\mathbb{R}$. Int Math Res Notices, (2015), (12): 43224334. PDF document

$C^{1,\alpha}$regularity for surfaces with mean curvature in $L^p$. Ann. Global Anal. Geom. 46 (2014), no. 2, 159186. PDF document

Density estimates for compact surfaces with total boundary curvature less than $4\pi$. Comm. Partial Differential Equations 37 (2012), no. 10, 18701886. PDF document

Curvature estimates for surfaces with bounded mean curvature. Trans. Amer. Math. Soc. 364 (2012), no. 11, 58135828. PDF document

The number of constant mean curvature isometric immersions of a surface. Comment. Math. Helv. 88 (2013), no. 1, 163183. PDF document

The rigidity of embedded constant mean curvature surfaces. J. Reine Angew. Math. 660 (2011), 181190. PDF document

Existence of regular neighborhoods for $H$surfaces. Illinois J. Math. 55, no. 3, 835844 (2011). PDF document

On curvature estimates for constant mean curvature surfaces. Geometric analysis: partial differential equations and surfaces, 165185, Contemp. Math., 570, Amer. Math. Soc., Providence, RI, 2012. PDF document

The Dynamics Theorem for CMC surfaces in $\mathbb{R}^3$. J. Differential Geometry, 85 (2010), 141173. PDF document

On the moduli space of constant mean curvature isometric immersions of a surface. Seminari di Geometria 20052009, Università di Bologna: 127136, 2010. PDF document

Curvature bounds for minimal surfaces with total boundary curvature less than $4\pi$. Proc. Amer. Maths. Soc, (2009), 24452450. PDF document

Structure theorems for embedded disks with mean curvature bounded in $L^p$. Comm. Anal. Geom. 16 (2008), no. 4, 819836. PDF document

Multivalued graphs in embedded constant mean curvature disks. Trans. Amer. Math. Soc., 359:143164, 2007. PDF document

A generalization of Rado's theorem for almost graphical boundaries. Math. Zeit., 251:849858, 2005. PDF document

Local behavior of embedded constant mean curvature disks. Seminari di Geometria 20012004, Università di Bologna: 7380, 2005. PDF document