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John McCuan

Bio: John McCuan is an academic researcher from Georgia Institute of Technology. The author has contributed to research in topics: Mean curvature & Constant-mean-curvature surface. The author has an hindex of 10, co-authored 29 publications receiving 268 citations. Previous affiliations of John McCuan include Max Planck Society & University of California, Berkeley.

Papers
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Journal ArticleDOI
TL;DR: In this paper, it was shown that the Alexandrov planar reflection problem admits no solution for embedded ring-type surfaces with two boundary components and Euler-Poincare characteristic zero.
Abstract: We consider embedded ring-type surfaces (that is, compact, connected, orientable surfaces with two boundary components and Euler-Poincare characteristic zero) in ${\bold R}^3$ of constant mean curvature which meet planes $\Pi_1$ and $\Pi_2$ in constant contact angles $\gamma_1$ and $\gamma_2$ and bound, together with those planes, an open set in ${\bold R}^3$. If the planes are parallel, then it is known that any contact angles may be realized by infinitely many such surfaces given explicitly in terms of elliptic integrals. If $\Pi_1$ meets $\Pi_2$ in an angle $\alpha$ and if $\gamma_1+\gamma_2>\pi+\alpha$, then portions of spheres provide (explicit) solutions. In the present work it is shown that if $\gamma_1+\gamma_2\le\pi+\alpha$, then the problem admits no solution. The result contrasts with recent work of H.C.~Wente who constructed, in the particular case $\gamma_1 = \gamma_2 =\pi/2$, a {\it self-intersecting} surface spanning a wedge as described above. Our proof is based on an extension of the Alexandrov planar reflection procedure to a reflection about spheres, on the intrinsic geometry of the surface, and on a new maximum principle related to surface geometry. The method should be of interest also in connection with other problems arising in the global differential geometry of surfaces.

38 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that with the exception of very particular cases, any tubular liquid bridge con guration joining parallel plates in the absence of gravity must change continuously with tilting of the plates thereby proving a conjecture of Concus and Finn Phys Fluids.
Abstract: It is shown that with the exception of very particular cases any tubular liquid bridge con guration joining parallel plates in the absence of gravity must change dis continuously with tilting of the plates thereby proving a conjecture of Concus and Finn Phys Fluids Thus the stability criteria that have appeared previously in the literature which take no account of such tilting are to some extent misleading Conceivable con gurations of the liquid mass following a plate tilting are characterized and conditions are presented under which stable drops in wedges with disk type or tubular free bounding surfaces can be expected As a corollary of the study a new existence theorem for H graphs over a square with discontinuous data is obtained The resulting surfaces can be interpreted as generalizations of the Scherk minimal surface in two senses a the requirement of zero mean curvature is weakened to constant mean curvature and b the boundary data of the Scherk surface which al ternate between the constants and on adjacent sides of a square are replaced by capillary data alternating between two constant values restricted by a geometrical criterion

34 citations

Journal ArticleDOI
TL;DR: In this article, first order necessary conditions for capillary surfaces in equilibrium in contact with solid surfaces which may also be allowed to move are derived for a floating ball in a fixed container of liquid.
Abstract: Well known first order necessary conditions for a liquid mass to be in equilibrium in contact with a fixed solid surface declare that the free surface interface has mean curvature prescribed in terms of the bulk accelerations acting on the liquid and meets the solid surface in a materially dependent contact angle. We derive first order necessary conditions for capillary surfaces in equilibrium in contact with solid surfaces which may also be allowed to move. These conditions consist of the same prescribed mean curvature equation for the interface, the same prescribed contact angle condition on the boundary, and an additional integral condition which may be said to involve, somewhat surprisingly, only the wetted region. An example of the kind of system under consideration is that of a floating ball in a fixed container of liquid. We apply our first order conditions to this particular problem.

24 citations

Journal ArticleDOI
TL;DR: In this article, the regularity of a capillary graph over a corner domain of angle α was investigated and an explicit construction of minimal surface solutions was given, which was later extended to the case of a cylindrical tube.
Abstract: This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions.

18 citations


Cited by
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Journal ArticleDOI
01 Apr 1899-Nature
TL;DR: In this paper, the authors pointed out that the distinction between "finite" and "infinite" is one which does not require definition, and that the authors' view is not the only accepted view.
Abstract: THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

740 citations

Journal ArticleDOI
30 Apr 2014
TL;DR: In this article, the authors review part of the classical theory of curves and surfaces in 3D Lorentz-Minkowski space and focus in spacelike surfaces with constant mean curvature pointing the differences and similarities with the Euclidean space.
Abstract: We review part of the classical theory of curves and surfaces in 3-dimensional Lorentz-Minkowski space. We focus in spacelike surfaces with constant mean curvature pointing the differences and similarities with the Euclidean space.

175 citations

Book ChapterDOI
01 Jan 1986
TL;DR: In this paper, it is taken as axiomatic that the equations of mathematical physics are invariant with respect to the action of a Lie group on tensor fields over a manifold.
Abstract: The equations of mathematical physics are typically ordinary or partial differential equations for vector or tensor fields over Riemannian manifolds whose group of isometries is a Lie group. It is taken as axiomatic that the equations be independent of the observer, in a sense we shall make precise below; and the consequence of this axiom is that the equations are invariant with respect to the group action. The action of a Lie group on tensor fields over a manifold is thus of primary importance. The action of a Lie group on a manifold M induces in a natural way automorphisms of the algebra of C∞ functions over M and on the algebra of tensor fields over M. The one parameter subgroups of the group induce one parameter subgroups of automorphisms of the tensor fields. The infinitesimal generators of these groups of automorphisms are the Lie derivatives of the action.

162 citations

Journal ArticleDOI
TL;DR: In this paper, a new approach for non-existence of positive solutions of cooperative semilinear elliptic systems with the Laplacian as principal part is introduced, based upon a new development of the method of moving spheres.

136 citations