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 least-area 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.

## 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.

1. W. H. Meeks III and G. Tinaglia. One-sided curvature estimates for H-disks. To appear in Cambridge Journal of Mathematics PDF document
2. T. Bourni, M. Langford and G. Tinaglia. A collapsing ancient solution of mean curvature flow in $\mathbb{R}^3$. To appear in J. Differential Geometry PDF document
3. W. H. Meeks III and G. Tinaglia. Curvature estimates for constant mean curvature surfaces. Duke Math. J. Volume 168, Number 16 (2019), 3057-3102. PDF document
4. T. Bourni, M. Langford and G. Tinaglia. Convex ancient solutions to curve shortening flow. Calc. Var. 59, 133 (2020). PDF document
5. T. Bourni, M. Langford and G. Tinaglia. On the existence of translating solutions of mean curvature flow in slab regions. Anal. PDE Volume 13, Number 4 (2020), 1051-1072. PDF document
6. T. Bourni, M. Langford and G. Tinaglia. Convex ancient solutions to mean curvature flow. To appear in Proceedings of the Australian-German Workshop on Differential Geometry in the Large PDF document
7. W. H. Meeks III and G. Tinaglia. Limit lamination theorem for H-surfaces. Journal für die reine und angewandte Mathematik, Volume 2019, Issue 748, Pages 269–296 PDF document
8. T. Bourni, M. Langford and G. Tinaglia. Translating solutions to mean curvature flow. To appear in Proceedings of the workshops on “Minimal surfaces: integrable systems and visualisation” PDF document
9. T. Bourni, M. Langford and G. Tinaglia. Ancient solutions to mean curvature flow. To appear in Proceedings of the FCGM-2018 PDF document
10. W. H. Meeks III and G. Tinaglia. Triply periodic constant mean curvature surfaces. Advances in Mathematics, Volume 335, 7 September 2018, Pages 809-837. PDF document
11. W. H. Meeks III, J. Pérez and G. Tinaglia. Constant mean curvature surfaces. Surveys in Differential Geometry, International Press, (2016). PDF document
12. W. H. Meeks III and G. Tinaglia. Chord arc properties for constant mean curvature disks. Geometry & Topology 22 (2018) 305–322. PDF document
13. B. Coskunuzer, W. H. Meeks III and G. Tinaglia. Non-properly embedded H-planes in $\mathbb{H}^2\times\mathbb{R}$. Math. Ann. (2018), 370(3), 1491-1512. PDF document
14. B. Coskunuzer, W. H. Meeks III and G. Tinaglia. Non-properly embedded H-planes in $\mathbb{H}^3$. J. Differential Geometry, Volume 105, Number 3 (2017), 405-425. PDF document
15. J. Bernstein and G. Tinaglia. Topological Type of Limit Laminations of Embedded Minimal Disks. J. Differential Geometry, Volume 102, Number 1 (2016), 1-23. PDF document
16. M. M. Rodríguez and G. Tinaglia. Non-proper complete minimal surfaces embedded in $\mathbb{H}^2\times\mathbb{R}$. Int Math Res Notices, (2015), (12): 4322-4334. PDF document
17. T. Bourni and G. Tinaglia. $C^{1,\alpha}$-regularity for surfaces with mean curvature in $L^p$. Ann. Global Anal. Geom. 46 (2014), no. 2, 159-186. PDF document
18. T. Bourni and G. Tinaglia. Density estimates for compact surfaces with total boundary curvature less than $4\pi$. Comm. Partial Differential Equations 37 (2012), no. 10, 1870-1886. PDF document
19. T. Bourni and G. Tinaglia. Curvature estimates for surfaces with bounded mean curvature. Trans. Amer. Math. Soc. 364 (2012), no. 11, 5813-5828. PDF document
20. B. Smyth and G. Tinaglia. The number of constant mean curvature isometric immersions of a surface. Comment. Math. Helv. 88 (2013), no. 1, 163-183. PDF document
21. G. Tinaglia. Review of: A course in minimal surfaces (Graduate Studies in Mathematics 121) By Tobias Holck Colding and William P. Minicozzi II. Bull. London Math. Soc. (2012) 44(2): 406-408. PDF document
22. W. H. Meeks III and G. Tinaglia. The rigidity of embedded constant mean curvature surfaces. J. Reine Angew. Math. 660 (2011), 181-190. PDF document
23. W. H. Meeks III and G. Tinaglia. Existence of regular neighborhoods for H-surfaces. Illinois J. Math. 55, no. 3, 835-844 (2011). PDF document
24. G. Tinaglia. On curvature estimates for constant mean curvature surfaces. Geometric analysis: partial differential equations and surfaces, 165-185, Contemp. Math., 570, Amer. Math. Soc., Providence, RI, 2012. PDF document
25. W. H. Meeks III and G. Tinaglia. The Dynamics Theorem for CMC surfaces in $\mathbb{R}^3$. J. Differential Geometry, 85 (2010), 141-173. PDF document
26. G. Tinaglia. On the moduli space of constant mean curvature isometric immersions of a surface. Seminari di Geometria 2005-2009, Università di Bologna: 127-136, 2010. PDF document
27. G. Tinaglia. Curvature bounds for minimal surfaces with total boundary curvature less than $4\pi$. Proc. Amer. Maths. Soc, (2009), 2445-2450. PDF document
28. G. Tinaglia. Structure theorems for embedded disks with mean curvature bounded in $L^p$. Comm. Anal. Geom. 16 (2008), no. 4, 819-836. PDF document
29. G. Tinaglia. Multi-valued graphs in embedded constant mean curvature disks. Trans. Amer. Math. Soc., 359:143-164, 2007. PDF document
30. B. Dean and G. Tinaglia. A generalization of Rado's theorem for almost graphical boundaries. Math. Zeit., 251:849-858, 2005. PDF document
31. G. Tinaglia. Local behavior of embedded constant mean curvature disks. Seminari di Geometria 2001-2004, Università di Bologna: 73-80, 2005. PDF document

## Preprints

T. Bourni, M. Langford and G. Tinaglia. The atomic structure of ancient grain boundaries. PDF document
W. H. Meeks III and G. Tinaglia. The geometry of constant mean curvature surfaces in $\mathbb{R}^3$. PDF document
W. H. Meeks III and G. Tinaglia. Limit lamination theorem for H-disks. PDF document