49 citations · 106 across the 18 of their papers we have counts for
22 papers
A convergence result for Mean Curvature Flow of totally real submanifolds
Tristan C. Collins, Adam Jacob, Yu-Shen Lin
We establish a convergence result for the mean curvature flow starting from a totally real submanifold which is "almost minimal" in a precise, quantitative sense. This extends, and…
Singularity formation along the line bundle mean curvature flow
Yu Hin Chan, Adam Jacob
The line bundle mean curvature flow is a complex analogue of the mean curvature flow for Lagrangian graphs, with fixed points solving the deformed Hermitian-Yang-Mills equation. In…
The Deformed Hermitian-Yang-Mills Equation and Level Sets of Harmonic Polynomials
Adam Jacob
Suppose is an entire harmonic polynomial with no critical points in the right half plane. Let lie on a level set of $…
The Torelli Theorem for Gravitational Instantons
Tristan Collins, Adam Jacob, Yu-Shen Lin
We give a short proof of the Torelli theorem for gravitational instantons, using the authors' previous construction of mirror special Lagrangian fibrations in del Pezzo sur…
The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces
Tristan C. Collins, Adam Jacob, Yu-Shen Lin
We prove a version of the Strominger-Yau-Zaslow mirror symmetry conjecture for non-compact Calabi-Yau surfaces arising from, on the one hand, pairs of a del…
The deformed Hermitian-Yang-Mills equation on the blowup of
Adam Jacob, Norman Sheu
We study the deformed Hermitian-Yang-Mills equation on the blowup of complex projective space. Using symmetry, we express the equation as an ODE which can be solved using combinato…
Weak Geodesics for the deformed Hermitian-Yang-Mills equation
Adam Jacob
We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowi…
Special Lagrangian submanifolds of log Calabi-Yau manifolds
Tristan C. Collins, Adam Jacob, Yu-Shen Lin
We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian-Yau. We prove that if…
Adiabatic limits of anti-self-dual connections on collapsed K3 surfaces
Ved Datar, Adam Jacob, Yuguang Zhang
We prove a convergence result for a family of Yang-Mills connections over an elliptic surface as the fibers collapse. In particular, assume is projective, admits a sec…
Hermitian-Yang-Mills connections on collapsing elliptically fibered surfaces
Ved Datar, Adam Jacob
Let be an elliptically fibered surface, admitting a sequence of Ricci-flat metrics collapsing the fibers. Let be a holomorphic …
Tangent cones of Hermitian Yang-Mills connections with isolated singularities
Adam Jacob, Henrique Sá Earp, Thomas Walpuski
We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on $…
Hermitian Yang-Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds
Adam Jacob, Thomas Walpuski
We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical Kähler manifolds: If is a reflexive sheaf over an ACyl Kähler manifold, whi…
(1,1) forms with specified Lagrangian phase: A priori estimates and algebraic obstructions
Tristan C. Collins, Adam Jacob, Shing-Tung Yau
Let be a Kähler manifold of dimension n, and let . We study the problem of specifying the Lagrangian phase of with respect to , which…
A special Lagrangian type equation for holomorphic line bundles
Adam Jacob, Shing-Tung Yau
Let be a holomorphic line bundle over a compact Kähler manifold . Motivated by mirror symmetry, we study the deformed Hermitian-Yang-Mills equation on , which is the line…
Poisson metrics on flat vector bundles over non-compact curves
Tristan C. Collins, Adam Jacob, Shing-Tung Yau
Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the…
Remarks on the Yang-Mills flow on a compact Kahler manifold
Tristan C. Collins, Adam Jacob
We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X. We construct a natural barrier function along the flow, and introduce some technique…
Existence of approximate Hermitian-Einstein structures on semistable principal bundles
Indranil Biswas, Adam Jacob, Matthias Stemmler
Let E_G be a principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and onl…
On the convergence of the Sasaki-Ricci flow
Tristan C. Collins, Adam Jacob
Given a Sasaki manifold S, we prove the Sasaki-Ricci flow converges exponentially fast to a Sasaki-Einstein metric if one exists, provided the automorphism group of the transverse…
Stable Higgs bundles and Hermitian-Einstein metrics on non-Kähler manifolds
Adam Jacob
Let be a compact Gauduchon manifold, and let and be holomorphic vector bundles over . Suppose that is stable when considering all subsheaves preserved by a Hig…
The Yang-Mills flow and the Atiyah-Bott formula on compact Kahler manifolds
Adam Jacob
We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X . Along a solution of the flow, we show the curvature approaches in …
The limit of the Yang-Mills flow on semi-stable bundles
Adam Jacob
By the work of Hong and Tian it is known that given a holomorphic vector bundle E over a compact Kahler manifold X, the Yang-Mills flow converges away from an analytic singular set…
Existence of approximate Hermitian-Einstein structures on semi-stable bundles
Adam Jacob
The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle E over a compact Kahler manifold X. It is shown that, if E is semi-stable, then Donalds…