# Compactness for manifolds and integral currents with bounded diameter and volume

## Related Articles

- Weak convergence of currents and cancellation. Sormani, Christina; Wenger, Stefan // Calculus of Variations & Partial Differential Equations;May2010, Vol. 38 Issue 1/2, p183
In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit...

- Strict p-negative type of a metric space. Hanfeng Li; Weston, Anthony // Positivity;Sep2010, Vol. 14 Issue 3, p529
Doust and Weston (J Funct Anal 254:2336â€“2364, 2008) have introduced a new method called enhanced negative type for calculating a non-trivial lower bound $${\wp_{T}}$$ on the supremal strict p-negative type of any given finite metric tree ( T, d). In the context of finite metric trees any...

- Canonical embeddings of compact metric spaces. Zatitskiy, P. // Journal of Mathematical Sciences;Apr2011, Vol. 174 Issue 1, p19
We prove that for any infinite compact metric space, two canonical embeddings (the Hausdorff-Kuratowski and Kantorovich-Rubinshtein embeddings) do not coincide. Bibliography: 3 titles.

- Parallel Vector Field Embedding. Binbin Lin; Xiaofei He; Chiyuan Zhang; Ming Ji // Journal of Machine Learning Research;Oct2013, Vol. 14, p2945
We propose a novel local isometry based dimensionality reduction method from the perspective of vector fields, which is called parallel vector field embedding (PFE). We first give a discussion on local isometry and global isometry to show the intrinsic connection between parallel vector fields...

- DERIVATIVES AND CURRENTS ON METRIC (MEASURE) SPACES. Gong, Jasun // Real Analysis Exchange;Aug2007 Summer Symposium, Vol. 32, p217
The article provides a summary of a presentation by Jasun Gong on derivatives and currents on metric spaces presented at the Thirty-First Summer Symposium in Real Analysis at the University of Oxford in England in August 2007. The paper aims to relate two theories in the setting of metric...

- Semicontinuity of eigenvalues under intrinsic flat convergence. Portegies, Jacobus // Calculus of Variations & Partial Differential Equations;Oct2015, Vol. 54 Issue 2, p1725
We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of...

- Growth of balls of holomorphic sections and energy at equilibrium. Berman, Robert; Boucksom, Sébastien // Inventiones Mathematicae;Aug2010, Vol. 181 Issue 2, p337
Let L be a big line bundle on a compact complex manifold X. Given a non-pluripolar compact subset K of X and a continuous Hermitian metric eâˆ’ Ï† on L, we define the energy at equilibrium of ( K, Ï†) as the Monge-AmpÃ¨re energy of the extremal psh weight associated to ( K, Ï†)....

- Hofer's diameter and Lagrangian intersections. Polterovich, Leonid // IMRN: International Mathematics Research Notices;1998, Vol. 1998 Issue 4, p217
The article presents a study on the infinite diameter of the 2-sphere Hamiltonian diffeomorphisms relative to Hofer's metrics. It mentions the energy-capacity inequality of high genus surfaces and aspherical symplectic manifolds from a link between the action spectrum and Hofer's norm. It also...

- The structure of 3-manifolds of complexity zero. Matveev, S.; Nikolaev, D. // Doklady Mathematics;Mar2014, Vol. 89 Issue 2, p143
The article offers information on the structure of three-manifolds which is considered as two-dimensional. It discusses complexity of a manifold M which is defined as the number c(M) of the true vertices of its minimal almost simple spine. It mentions that a spine is simple when link of each of...