Abstract: We show that for an axially symmetric anisotropic surface energy, only stable disc-type surfaces with constant anisotropic mean curvature bounded by a circle which lies in a plane orthogonal to the rotation axis of the Wulff shape are rescalings of parts of the Wulff shape and the flat disc. References [Enhancements On Off] What's this? A level set approach. MR  J. Cahn, D. Hoffman, A vector thermodynamics for anisotropic surfaces-II. Curved and faceted surfaces, Acta Metallurgica, Volume 22, Issue 10 , Notes taken by Peter Lax and John W.
Gray; With a preface by S. Chern; With a preface by K. Differential Geom. More generally, a topological surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the upper half-plane H 2 in C. These homeomorphisms are also known as coordinate charts. The boundary of the upper half-plane is the x -axis. A point on the surface mapped via a chart to the x -axis is termed a boundary point.
ISBN 10: 3642396259
The collection of such points is known as the boundary of the surface which is necessarily a one-manifold, that is, the union of closed curves. On the other hand, a point mapped to above the x -axis is an interior point. The collection of interior points is the interior of the surface which is always non- empty. The closed disk is a simple example of a surface with boundary.
Surface (topology) - Wikipedia
The boundary of the disc is a circle. The term surface used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense.
A surface with empty boundary which is compact is known as a 'closed' surface. The two-dimensional sphere, the two-dimensional torus , and the real projective plane are examples of closed surfaces. In differential and algebraic geometry , extra structure is added upon the topology of the surface. This added structures can be a smoothness structure making it possible to define differentiable maps to and from the surface , a Riemannian metric making it possible to define length and angles on the surface , a complex structure making it possible to define holomorphic maps to and from the surface—in which case the surface is called a Riemann surface , or an algebraic structure making it possible to detect singularities , such as self-intersections and cusps, that cannot be described solely in terms of the underlying topology.
Historically, surfaces were initially defined as subspaces of Euclidean spaces. Often, these surfaces were the locus of zeros of certain functions, usually polynomial functions. Such a definition considered the surface as part of a larger Euclidean space, and as such was termed extrinsic. In the previous section, a surface is defined as a topological space with certain properties, namely Hausdorff and locally Euclidean.
This topological space is not considered a subspace of another space. In this sense, the definition given above, which is the definition that mathematicians use at present, is intrinsic. A surface defined as intrinsic is not required to satisfy the added constraint of being a subspace of Euclidean space. It may seem possible for some surfaces defined intrinsically to not be surfaces in the extrinsic sense.
On the Shape of Compact Hypersurfaces with Almost‐Constant Mean Curvature
However, the Whitney embedding theorem asserts every surface can in fact be embedded homeomorphically into Euclidean space, in fact into E 4 : The extrinsic and intrinsic approaches turn out to be equivalent. In fact, any compact surface that is either orientable or has a boundary can be embedded in E 3 ; on the other hand, the real projective plane, which is compact, non-orientable and without boundary, cannot be embedded into E 3 see Gramain. Steiner surfaces , including Boy's surface , the Roman surface and the cross-cap , are models of the real projective plane in E 3 , but only the Boy surface is an immersed surface.
All these models are singular at points where they intersect themselves. The Alexander horned sphere is a well-known pathological embedding of the two-sphere into the three-sphere. The chosen embedding if any of a surface into another space is regarded as extrinsic information; it is not essential to the surface itself. For example, a torus can be embedded into E 3 in the "standard" manner which looks like a bagel or in a knotted manner see figure.
The two embedded tori are homeomorphic, but not isotopic : They are topologically equivalent, but their embeddings are not. The image of a continuous, injective function from R 2 to higher-dimensional R n is said to be a parametric surface. Such an image is so-called because the x - and y - directions of the domain R 2 are 2 variables that parametrize the image. A parametric surface need not be a topological surface. A surface of revolution can be viewed as a special kind of parametric surface.
If f is a smooth function from R 3 to R whose gradient is nowhere zero, then the locus of zeros of f does define a surface, known as an implicit surface. If the condition of non-vanishing gradient is dropped, then the zero locus may develop singularities. Each closed surface can be constructed from an oriented polygon with an even number of sides, called a fundamental polygon of the surface, by pairwise identification of its edges.
- Bob: No Ordinary Cat.
- Constant Mean Curvature Surfaces with Boundary (Springer!
- Recommended for you.
- Constant Mean Curvature Surfaces with Boundary (Springer - Fresno First & CPR Trai Lib.
- Constant Mean Curvature Surfaces with Boundary | Rafael López | Springer;
For example, in each polygon below, attaching the sides with matching labels A with A , B with B , so that the arrows point in the same direction, yields the indicated surface. Any fundamental polygon can be written symbolically as follows. Begin at any vertex, and proceed around the perimeter of the polygon in either direction until returning to the starting vertex.
During this traversal, record the label on each edge in order, with an exponent of -1 if the edge points opposite to the direction of traversal. The four models above, when traversed clockwise starting at the upper left, yield. Note that the sphere and the projective plane can both be realized as quotients of the 2-gon, while the torus and Klein bottle require a 4-gon square.
The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon edge labels as generators. This is a consequence of the Seifert—van Kampen theorem. Gluing edges of polygons is a special kind of quotient space process.
The quotient concept can be applied in greater generality to produce new or alternative constructions of surfaces. For example, the real projective plane can be obtained as the quotient of the sphere by identifying all pairs of opposite points on the sphere. Another example of a quotient is the connected sum. The connected sum of two surfaces M and N , denoted M N , is obtained by removing a disk from each of them and gluing them along the boundary components that result.
The boundary of a disk is a circle, so these boundary components are circles. This is because deleting a disk from the sphere leaves a disk, which simply replaces the disk deleted from M upon gluing. Connected summation with the torus T is also described as attaching a "handle" to the other summand M. If M is orientable, then so is T M. The connected sum is associative, so the connected sum of a finite collection of surfaces is well-defined.
The connected sum of two real projective planes, P P , is the Klein bottle K. Thus, the connected sum of three real projective planes is homeomorphic to the connected sum of the real projective plane with the torus. Any connected sum involving a real projective plane is nonorientable.
A closed surface is a surface that is compact and without boundary. Examples are spaces like the sphere , the torus and the Klein bottle. The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families:. The surfaces in the first two families are orientable. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number g of tori involved is called the genus of the surface. The surfaces in the third family are nonorientable.
It follows that a closed surface is determined, up to homeomorphism, by two pieces of information: its Euler characteristic, and whether it is orientable or not. In other words, Euler characteristic and orientability completely classify closed surfaces up to homeomorphism. Closed surfaces with multiple connected components are classified by the class of each of their connected components, and thus one generally assumes that the surface is connected.
Relating this classification to connected sums, the closed surfaces up to homeomorphism form a commutative monoid under the operation of connected sum, as indeed do manifolds of any fixed dimension. This relation is sometimes known as Dyck's theorem after Walther von Dyck , who proved it in Dyck , and the triple cross surface P P P is accordingly called Dyck's surface.
Related Constant Mean Curvature Surfaces with Boundary (Springer Monographs in Mathematics)
Copyright 2019 - All Right Reserved