Bubble Tree Research Articles

ENERGY CONCENTRATION FOR 2-DIMENSIONAL RADIALLY SYMMETRIC EQUIVARIANT HARMONIC MAP HEAT FLOWS

We give a description of singularity formation in terms of energy quanta for 2-dimensional radially symmetric equivariant harmonic map heat flows. Adapting Struwe's energy method we first establish a finite bubble tree result with a discrete multiple of energy quanta disappearing in the singularity. We then use intersection-comparison arguments to show that the bubble tree consists of a single bubble only and that there is a well defined scale RBHK(t) ↓ 0 in which the solution converges to the standard harmonic map.

Harmonic maps from the Riemann sphere into the complex projective space and the harmonic sequences

When harmonic maps from the Riemann sphere into the complex projective space are energy bounded, it contains a subsequence converging to a bubble tree map fI: TI → CPn. We show that their ∂-transforms and $\overline{\partial}$-transforms are also energy bounded. Hence their subsequences converge to harmonic bubble tree maps $f_1^{I_1}:T^{I_1}$ → CPn and $f_{-1}^{I_{-1}}:T^{I_{-1}}$ → CPn respectively. In this paper, we show relations between fI, $f_1^{I_1}$ and $f_{-1}^{I_{-1}}$.

Bubble tree convergence for the harmonic sequence of harmonic surfaces in ℂℙ n

We show that any harmonic sequence determined by a harmonic map from a compact Riemannian surface M to ℂℙn has a terminating holomorphic (or anti-holomorphic) map from M to ℂℙn, or a “bubble tree limit” consisting of a harmonic map \( \hat f \): M → ℂℙn and a tree of bubbles hλµ: S2 → ℂℙn.

Smoothness on bubble tree compactified instanton moduli spaces

The bubble tree compactified instanton moduli space \( \bar M_\kappa \)(X) is introduced. Its singularity set Singκ(X) is described. By the standard gluing theory, one can show that \( \bar M_\kappa \)(X) — Singκ(X) is a topological orbifold. In this paper, we give an argument to construct smooth structures on it.

Some Progress in Conformal Geometry

This is a survey paper of our current research on the theory of partial differen- tial equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the 2-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.

On a Conformal Gap and Finiteness Theorem for a Class of Four-Manifolds

In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and make a comparison of the solutions of the σk equations on a degenerating family of Bach-flat metrics.

On the nonexistence of finite time bubble trees in symmetric harmonic map heat flows from the disk to the 2-sphere

Let B2⊂R2 and S2⊂R3 be the unit disk and the unit sphere, and let v:B2×R+→S2 be a radially symmetric harmonic map heat flow, whose singularities coincide with downward energy jumps. Then its finite time singularities are simple in the sense that precisely one harmonic sphere separates at a time.

Bubbling of the heat flows for harmonic maps from surfaces

In this article we prove that any Palais-Smale sequence of the energy functional on surfaces with uniformly L2-bounded tension fields converges pointwise, by taking a subsequence if necessary, to a map from connected (possibly singular) surfaces, which consist of the original surfaces and finitely many bubble trees. We therefore get the corresponding results about how the solutions of heat flow for harmonic maps from surfaces form singularities at infinite time. © 1997 John Wiley & Sons, Inc.

Bubble tree convergence for harmonic maps

Let Σ be a compact Riemann surface. Any sequence fn : Σ — > M of harmonic maps with bounded energy has a tree consisting of a harmonic map /o : Σ -> M and a tree of bubbles fk : S2 -> M. We give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class, and that the images of the fn converge pointwise. We then give explicit counterexamples showing that bubble tree convergence fails (i) for harmonic maps fn when the conformal structure of Σ varies with n, and (ii) when the conformal structure is fixed and {/n} is a Palais-Smale sequence for the harmonic map energy.

Pseudo-holomorphic maps and bubble trees

This paper proves a strong convergence theorem for sequences of pseudo-holomorphic maps from a Riemann surface to a symplectic manifoldN with tamed almost complex structure. (These are the objects used by Gromov to define his symplectic invariants.) The paper begins by developing some analytic facts about such maps, including a simple new isoperimetric inequality and a new removable singularity theorem. The main technique is a general procedure for renormalizing sequences of maps to obtain “bubbles on bubbles.” This is a significant step beyond the standard renormalization procedure of Sacks and Uhlenbeck. The renormalized maps give rise to a sequence of maps from a “bubble tree”—a map from a wedge Σ V S2 V S2 V ... →N. The main result is that the images of these renormalized maps converge in L1,2 ∪C° to the image of a limiting pseudo-holomorphic map from the bubble tree. This implies several important properties of the bubble tree. In particular, the images of consecutive bubbles in the bubble tree intersect, and if a sequence of maps represents a homology class then the limiting map represents this class. While the main focus is on holomorphic maps, the bubble tree construction applies to other conformally invariant problems, including minimal surfaces and Yang-Mills fields.

Software Reviews : Diagram-Master 5.0

This companion to Chart-Master produces diagrams useful to the social scientist for overheads and publication-quality graphics, including organization charts, GANTT and other time or schedule charts, flow charts, Venn logic diagrams or other bubble charts, tree diagrams, grid charts, matrices, pyramids, and freeform art work. A library of 101 frequently used shapes is included. Design- Master is an easy-to-use program allowing the user to build the diagram directly on the screen by answering menu prompts. (Promptscan even be answered by an optional light pen.) Diagram-Master supports a wide variety of printers and plotters as well as the Polaroid Palette in-line camera.