By using the conformal method, solutions of the Einstein-scalar ficld gravitational constraint equations are obtained. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwe...By using the conformal method, solutions of the Einstein-scalar ficld gravitational constraint equations are obtained. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. The proofs are constructive and allow for arbitrary dimension （〉 2） as well as low regularity initial data.展开更多
In the theory of general relativity, the finding of the Einstein Field Equation happens in a complex mathematical operation, a process we don’t need any more. Through a new theory in vector analysis, we’ll see that ...In the theory of general relativity, the finding of the Einstein Field Equation happens in a complex mathematical operation, a process we don’t need any more. Through a new theory in vector analysis, we’ll see that we can calculate the components of the Ricci tensor, Ricci scalar, and Einstein Field Equation directly in an easy way without the need to use general relativity theory hypotheses, principles, and symbols. Formulating the general relativity theory through another theory will make it easier to understand this relativity theory and will help combining it with electromagnetic theory and quantum mechanics easily.展开更多
In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota＇s argument we obtain a local lower bound estimate of ...In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota＇s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct （elliptic） proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.展开更多
基金Project supported by NSF Grant PHY-0354659 at the University of Oregonby NSF Grant DMS-0305048 at the University of Washington.
文摘By using the conformal method, solutions of the Einstein-scalar ficld gravitational constraint equations are obtained. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. The proofs are constructive and allow for arbitrary dimension （〉 2） as well as low regularity initial data.
文摘In the theory of general relativity, the finding of the Einstein Field Equation happens in a complex mathematical operation, a process we don’t need any more. Through a new theory in vector analysis, we’ll see that we can calculate the components of the Ricci tensor, Ricci scalar, and Einstein Field Equation directly in an easy way without the need to use general relativity theory hypotheses, principles, and symbols. Formulating the general relativity theory through another theory will make it easier to understand this relativity theory and will help combining it with electromagnetic theory and quantum mechanics easily.
文摘In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota＇s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct （elliptic） proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.
