K. Uhlenbeck's research while affiliated with University of Illinois at Chicago and other places
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Publications (20)
AssumeF is the curvature (field) of a connection (potential) onR
4 with finiteL
2 norm\(\left( {\int\limits_{R^4 } {\left| F \right|^2 dx< \infty } } \right)\). We show the chern number\(c_2= {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-\nulldelimiterspace} 8}\pi ^2 \int\limits_{R^4 } {F \wedge} F\) (topological quantum number) is an integer....
Most of E. Noether’s mathematical research was in the field of algebra, or related to it. Outside this work in algebra is one very famous theorem of Noether’s which is stated in every book on classical mechanics, as well as in many texts on more recently developed physical theories [20]. This well-known theorem applies to the following situation: a...
We show by means of the implicit function theorem that Coulomb gauges exist for fields over a ball inR
n
when the integralL
n/2 field norm is sufficiently small. We then are able to prove a weak compactness theorem for fields on compact manifolds withL
p
integral norms bounded,p>n/2.
We show that a field satisfying the Yang-Mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite. We obtain the result that every Yang-Mills field overR
4 with bounded functional (L
2 norm) may be obtained from a field onS
4=R
4{}. Hodge (or Coulomb) gauges are constructed for genera...
Let $M$ be a closed orientable surface of genus larger than zero and $N$ a compact Riemannian manifold. If $u: M \rightarrow N$ is a continuous map, such that the map induced by it between the fundamental groups of $M$ and $N$ contains no nontrivial element represented by a simple closed curve in its kernel, then there exists a conformal branched m...
There are many interesting variational problems for which the PalaisSmale condition cannot be verified. In cases where the Palais-Smale condition can be verified for an approximating integral, and the critical points converge, a Morse theory is valid. This theory applies to a class of variational problems consisting of the energy integral for harm...
In this paper we develop an existence theory for minimal 2-spheres in compact Riemannian manifolds. The spheres we obtain are conformally immersed minimal surfaces except at a finite number of isolated points, where the structure is that of a branch point. We obtain an existence theory for harmonic maps of orientable surfaces into Riemannian manifo...
Citations
... If u i : Σ → (X, g, I) is a sequence of holomorphic maps such with points p i ∈ u i (Σ) and p i → p, then by Proposition 5.3, the holomorphic curves u i (Σ) all lie in a fixed compact set. Since every curve in M 1 has the same volume and hence energy, the standard Gromov-Sacks-Uhlenbeck compactness theory [67,96,38,74] implies that u i converges to a cusp curve (or stable curve) u : ∪ α Σ α → (X, g, I). Here ∪ α Σ α is some tree of Riemann surfaces; its precise structure is irrelevant for our current considerations. ...
... A, where . A is the field in (9) for some orthogonal matrix Q (see [5,17,22]). ...
... The Uhlenbeck gauge was introduced in [46], the application to harmonic maps is due to Rivière [34]. See also [38] for a variational method, [15, Theorem 1.2] for n ≥ 3. ...
... The Hölder regularity of a minimizer u of F follows then from standard arguments (see [11], [8], [12] for instance). We infer from Lemma 5 that given B 2R (a) ⊂ Ω and u a minimizer of F , there exists β ∈ (0, 1), depending on N and N only, and C, depending on R, N , N , V and Ω |u| 2 , such that for 0 < r ≤ R, ˜ ...
... Generic spectral properties for metric graphs were first studied by Friedlander in 2005 [14]. Motivated by the Sturm-Liouville theory on intervals and the genericity works of Albert [2] and Uhlenbeck [29] on compact manifolds, Friedlander showed that for any graph 1 Γ of N edges, and a generic ℓ ∈ R N + , all eigenvalues of (Γ, ℓ) are simple. The next result was due to Berkolaiko and Liu [10] in 2017. ...
... In the local case, simplicity of the eigenvalues with respect to a perturbation of the coefficients where proved in [11] and we are able to show the nonlocal counterpart of this result. In particular, we prove that all the eigenvalues of (1.2) and (1.3) are simple for generic functions a and α, respectively, in this two results. ...
... Minimizers (or almost minimizers) of E 0 exhibit crucial compactness characteristics, as confirmed in [L] and [SU1], and these traits are fundamental in the examination of tangent maps. We will now present several lemmas that mirror these compactness outcomes for the Alt-Caffarelli energy. ...
... In this subsection, we summarize some important lemmas regarding harmonic maps. Here we follow [10] and [8]. Also, see [4] and [5] as a good summary of extensive results. ...
... The perturbation method has turned out to be a powerful tool in dealing with various variational problems (see, for example, [2,3,7,18,22,23,27,32,38,39,46,50,51]). The key point of this method is to find suitable perturbation schemes so that the perturbed problems provide additional information for solving the original problems. ...
... Let f : S → M be a π 1 -injective map of a hyperbolic surface S into M. Schoen-Yau [23] and Sacks-Uhlenbeck [20] show that f is homotopic to a minimal map f m . In addition, Uhlenbeck shows that if the principal curvatures ±λ(p) of f m (S) satisfy λ(p) ∈ (−1, 1) for every p ∈ f m (S), then f m is quasifuchsian and it is the unique minimal map in its homotopy class. ...
