Detecting deformities on objects is a typical topic in shape analysis, and has much applications such as abnormalities detection in medical imaging (e.g. growth of tumor, spread of cancer). While many algorithms are already well-established in a 2-dimensional case when the object is indeed a surface, a model that still performs well in the general n-dimensional case is still missing. It is our goal in this paper to complete this missing piece, by introducing an indicator in order to effectively distinguish between normal and abnormal deformities. The proposed framework is closely related to the classic 2-dimensional conformal geometry and quasi-conformal geometry. In this work, we model abnormal deformations by anisotropic deformations. Given any two objects of the same dimension (with landmark constraints in between), we define the Anisotropic Indicator, a locally defined real-valued function on the original object, which demonstrates the abnormalities in the deformation between them. Both global and local features about the abnormalities between the two objects can be tracked by analyzing the indicator. We tested the algorithm by detecting deformations on synthetic data and real data, and results show that our algorithm can detect deformations of different types and degrees.

In this paper, we study the space of translational limits T (M) of a surface M properly embedded in R3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface Σ ∈ T (M) the set T (Σ) ⊂ T(M). Among various dynamics type results we prove that surfaces in minimal T -invariant sets of T (M) are chord-arc. We also show that if M has an infinite number of ends, then there exists a nonempty minimal T -invariant set in T (M) consisting entirely of surfaces with planes of Alexandrov symmetry. Finally, when M has a plane of Alexandrov symmetry, we prove the following characterization theorem: M has finite topology if and only if M has a finite number of ends greater than one.

The holonomy of the ambient metrics of Nurowski’s conformal structures associated to generic real-analytic 2-plane fields on oriented 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form G2 of the exceptional Lie group, and is equal to G2 for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinitedimensional family of metrics of holonomy equal to split G2. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in G2 is established by proving an ambient extension theorem for parallel tractors in the context of conformal geometry in general signature and dimension, which is expected to be of independent interest.

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