Journal ArticleMathematische Annalen · April 1, 2025
We prove a uniqueness result for asymptotically conical (AC) gradient shrinking solitons for the Laplacian flow of closed G2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G2-cone, then their G2-structure ...
Full textCite
Journal Article · December 16, 2021
We initiate a systematic study of cohomogeneity-one solitons in Bryant's
Laplacian flow of closed G_2-structures on a 7-manifold, motivated by the
problem of understanding finite-time singularities of that flow. Here we focus
on solitons with symmetry grou ...
Link to itemCite
Journal ArticleDuke Mathematical Journal · October 15, 2021
Featured Publication
We develop a powerful new analytic method to construct complete noncompact Ricci-flat 7-manifolds, more specifically G2-manifolds, that is, Riemannian 7- manifolds .M;g/ whose holonomy group is the compact exceptional Lie group G2. Our construction gives t ...
Full textOpen AccessCite
Journal ArticleJournal of the European Mathematical Society · January 1, 2021
Featured Publication
We construct infinitely many new 1-parameter families of simply connected complete non-compact G2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the na ...
Full textCite
Journal ArticleAnnals of Mathematics · January 1, 2017
Featured Publication
There is a rich theory of so-called (strict) nearly Kahler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kahler 6-manifolds play a distinguished role both ...
Full textOpen AccessCite
Journal ArticleOberwolfach Reports · December 4, 2015
All currently known construction methods of smooth compact
\mathrm G_2
-manifolds have been tied to certain ...
Full textCite
Journal ArticleJournal of Differential Geometry · October 1, 2015
Let M be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to [0, ∞) x X for some compact connected Ricci-flat manifold X. We begin by proving general structure theorems for M; in particular we sho ...
Full textOpen AccessCite
Journal ArticleDuke Mathematical Journal · January 1, 2015
Featured Publication
We construct many new topological types of compact G2-manifolds, that is, Riemannian 7-manifolds with holonomy group G2. To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptot ...
Full textCite
Journal ArticleGeometry and Topology · July 15, 2013
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few h ...
Full textCite
Journal ArticleCommunications in Analysis and Geometry · January 1, 2013
SO(p) × SO(q)-invariant special Lagrangian cones in ℂp+q(equivalently, SO(p) × SO(q)-invariant special Legendrians in S2(p+q)-1) are an important family of special Lagrangians (SL) whose basic features were studied in our previous paper [13]. In some ways, ...
Full textCite
Journal ArticleCommunications in Analysis and Geometry · January 1, 2012
We study a construction we call the twisted product; in this construction higher dimensional special Lagrangian (SL) and Hamiltonian stationary cones in Cp+q (equivalently special Legendrian or contact stationary submanifolds in S2(p+q)?1) are constructed ...
Full textCite
Journal Article · May 9, 2010
We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL)
cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues of
the $\sorth{2}$-invariant SL cones constructed previously by MH and used in our
gluing constructions ...
Link to itemCite
Journal ArticleGeometry and Topology · October 21, 2006
We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/nonexistence of such smoothings of special Lagrangian cones is an important component of the cu ...
Full textCite
Journal ArticleInventiones Mathematicae · July 29, 2004
Featured Publication
We prove a number of results on the geometric complexity of special Lagrangian (SLG) T2-cones in ℂ3. Every SLG T 3-cone has a fundamental integer invariant, its spectral curve genus. We prove that the spectral curve genus of an SLG T2-cone gives a lower bo ...
Full textCite
Journal ArticleAmerican Journal of Mathematics · January 1, 2004
We study special Lagrangian cones in ℂn with isolated singularities especially the case n = 3. Our main result constructs an infinite family of special Lagrangian cones in ℂ3 each of which has a toroidal link. We obtain a detailed geometric description of ...
Full textCite
Journal ArticleJournal of Mathematical Physics · August 1, 2003
The most fruitful approach to studying low energy soliton dynamics in field theories of Bogomol'nyi type is the geodesic approximation of Manton. In the case of vortices and monopoles, Stuart has obtained rigorous estimates of the errors in this approximat ...
Full textCite
Journal ArticlePhysics Letters, Section A: General, Atomic and Solid State Physics · July 15, 2002
A systematic correlation between the initial profile of discrete breathers and their frequency is described. The context is that of a very weakly harmonically coupled chain of softly anharmonic oscillators. The results are structurally stable, that is, rob ...
Full textCite
Journal ArticleNonlinearity · November 1, 1998
Existence of breather (spatially localized, time periodic, oscillatory) solutions of the topological discrete sine-Gordon (TDSG) system, in the regime of weak coupling, is proved. The novelty of this result is that, unlike the systems previously considered ...
Full textCite
